The ability to quickly and correctly construct sentences in English allows ...
REFERENCE MATERIAL ON GEOMETRY FOR 7-11 CLASSES.
Dear Parents! If you are looking for a math tutor for your child, then this ad is for you. I offer skype tutoring: preparation for the OGE, the Unified State Exam, elimination of knowledge gaps. Your benefits are clear:
1) Your child is at home, and you can be calm for him;
2) Classes are held at a convenient time for the child, and you can even attend these classes. I explain in a simple and accessible way on all the familiar blackboard.
3) You can figure out other important advantages of skype classes yourself!
P.S. Friends, of course it's free!
Dear friends! Are you preparing for the OGE or the Unified State Exam?
To help you Geometry Reference 7-9 .
Definition of a parallelogram.
A parallelogram is a quadrilateral in which opposite sides are pairwise parallel: AB || CD, AD || DC.
Opposite sides of a parallelogram are equal: AB = CD, AD = DC.
The opposite angles of the parallelogram are:
∠A =∠C,∠B =∠D.
The sum of the angles of a parallelogram adjacent to one side of it is 180 °. For example, ∠ A +∠B = 180 °.
Any diagonal of a parallelogram divides it into two equal triangles. Δ ABD = Δ BCD.
The diagonals of the parallelogram intersect and the intersection point is halved. AO = OC, BO = OD. Let AC = d 1 and BD = d 2, ∠COD = α. The sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of all its sides:
- If two opposite sides of a quadrilateral are parallel and equal, then this quadrilateral is a parallelogram.
- If the opposite sides of a quadrilateral are pairwise equal, then this quadrilateral is a parallelogram.
- If the diagonals of a quadrilateral intersect and the intersection point is halved, then this quadrilateral is a parallelogram.
Parallelogram area.
1) S = ah;
2) S = ab ∙ sinα;
A rectangle is a parallelogram with all angles straight. ABCD- rectangle. A rectangle has all the properties of a parallelogram.
The diagonals of the rectangle are equal.
AC = BD. Let AC = d 1 and BD = d 2, ∠COD = α.
d 1 = d 2 - the diagonals of the rectangle are equal. α is the angle between the diagonals.
The square of the diagonal of a rectangle is equal to the sum of the squares of the sides of the rectangle:
(d 1) 2 = (d 2) 2 = a 2 + b 2.
Rectangle area can be found by the formulas:
1) S = ab; 2) S = (½) · d² ∙ sinα; (d is the diagonal of the rectangle).
A circle can be described around any rectangle, the center of which is the point of intersection of the diagonals; the diagonals are the diameters of the circle.
Rhombus.
A rhombus is a parallelogram in which all sides are equal.
ABCD- rhombus.
A rhombus has all the properties of a parallelogram.
The diagonals of the rhombus are mutually perpendicular.
AC | BD.
The diagonals of a rhombus are the bisectors of its corners.
The area of the rhombus.
1) S = ah;
2) S = a 2 ∙ sinα;
3) S = (½) d 1 ∙ d 2;
4) S = P ∙ r, where P is the perimeter of the rhombus, r is the radius of the inscribed circle.
Square.
All sides of the square are equal, the diagonals of the square are equal and intersect at right angles.
The diagonal of the square is d = a√2.
Square area. 1) S = a 2; 2) S = (½) d 2.
Trapezium.
Trapezoid bases AD || BC, MN-midline
Trapezium area equal to the product of the half-sum of its bases and the height:
S = (AD + BC) ∙ BF / 2 or S = (a + b) ∙ h / 2.
In an isosceles (isosceles) trapezoid, the lengths of the lateral sides are equal; the angles at the base are equal.
The area of any quadrangle.
- The area of any quadrangle is equal to half the product of its diagonals by the sine of the angle between them:
S = (½) d 1 ∙ d 2 ∙ sinβ.
- The area of any quadrangle is equal to half the product of its perimeter and the radius of the inscribed circle:
Inscribed and circumscribed quadrangles.
In a convex quadrilateral inscribed in a circle, the product of the diagonals is equal to the sum of the products of opposite sides (Ptolemy's theorem).
AC ∙ BD = AB ∙ DC + AD ∙ BC.
If the sums of the opposite angles of a quadrangle are 180 ° each, then a circle can be described around a quadrangle... The converse is also true.
If the sums of the opposite sides of a quadrilateral are equal (a + c = b + d), then a circle can be inscribed in this quadrangle. The converse is also true.
Circle, circle.
1) Circumference C = 2πr;
2) Area of a circle S = πr 2;
3) Arc length AB:
4) Area of the AOB sector:
5) Segment area (highlighted area):
(“-” is taken if α<180°; «+» берут, если α>180 °), ∠AOB = α - center angle. Arc l visible from the center O at an angle α.
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs: c² = a² + b².
Area of a right triangle.
S Δ= (½) a ∙ b, where a and b are legs or S Δ= (½) c ∙ h, where c is the hypotenuse, h is the height drawn to the hypotenuse.
The radius of a circle inscribed in a right-angled triangle.
Proportional line segments in a right-angled triangle.
The height drawn from the top of the right angle to the hypotenuse is the average proportional value between the projections of the legs to the hypotenuse: h 2 = a c ∙ b c;
and each leg is the average proportional value between the entire hypotenuse and the projection of this leg onto the hypotenuse: a 2 = c ∙ a c and b 2 = c ∙ b c ( the product of the middle members of the proportion is equal to the product of its extreme members: h, a, b - the middle terms of the corresponding proportions).
The sine theorem.
In any triangle, the sides are proportional to the sine of the opposite angles.
Corollary from the sine theorem.
Each of the ratios of the side to the sine of the opposite angle is 2R, where R Is the radius of a circle circumscribed about a triangle.
Cosine theorem.
The square of any side of a triangle is equal to the sum of the squares of its other two sides without twice the product of these sides by the cosine of the angle between them.
Properties of an isosceles triangle.
In an isosceles triangle ( side lengths are equal) the height drawn to the base is the median and the bisector. The angles at the base of an isosceles triangle are equal.
The sum of the interior angles of any triangle is 180 °, i.e. ∠1 + ∠2 + ∠3 = 180 °.
Outside corner of a triangle(∠4) is equal to the sum of two interior, not adjacent to it, that is, ∠4 = ∠1 + ∠2.
Midline of a triangle connects the midpoints of the sides of the triangle.
The midline of the triangle is parallel to the base and equal to half of it: MN = AC / 2.
Area of a triangle.
Heron's formula.
Center of gravity of the triangle.
The center of gravity of a triangle is the intersection of the medians, which divides each median by a 2: 1 ratio from the vertex.
Length of the median to side a:
The median divides a triangle into two equal triangles, the area of each of these two triangles being equal to half the area of this triangle.
Angle bisector of a triangle.
1) The bisector of the angle of any triangle divides the opposite side into parts, respectively proportional to the sides of the triangle:
2) if AD = β a, then the length of the bisector is:
3) All three bisectors of the triangle intersect at one point.
Center of a circle inscribed in a triangle, lies at the intersection of the bisectors of the angles of the triangle.
Area of a triangle S Δ = (½) P ∙ r, where P = a + b + c, r is the radius of the inscribed circle.
The radius of the inscribed circle can be found by the formula:
Center of a circle circumscribed about a triangle, lies at the intersection of the middle perpendiculars to 
sides of the triangle.
The radius of a circle around any triangle:
Radius of a circle circumscribed about a right-angled triangle, equal to half of the hypotenuse: R = AB / 2;
The medians of right-angled triangles drawn to the hypotenuse are equal to half of the hypotenuse (these are the radii of the circumscribed circle) OC = OC 1 = R.
Formulas for the radii of inscribed and circumscribed circles of regular polygons.
Circle, described near a regular n-gon.
Circle, inscribed into a regular n-gon.
The sum of the interior angles of any convex n-gon is 180 ° (n-2).
Sum of outside angles of any convex 0 n-gon is equal to 360 °.
Rectangular parallelepiped.
All faces of a rectangular parallelepiped are rectangles. a, b, c - linear dimensions of a rectangular parallelepiped (length, width, height).
1) Diagonal of a rectangular parallelepiped d 2 = a 2 + b 2 + c 2;
2) Lateral surface S side. = P main. ∙ H or S side. = 2 (a + b) c;
3) Full surface S full. = 2S main. + S side. or
S full = 2 (ab + ac + bc);
4) The volume of a rectangular parallelepiped V = S main. ∙ H or V = abc.
1) All the faces of the cube are squares with side a.
2) The diagonal of the cube d = a√3.
3) The side surface of the cube S side. = 4a 2;
4) The full surface of the cube S full. = 6a 2;
5) Volume of the cube V = a 3.
Straight parallelepiped(there is a parallelogram or rhombus at the base, the lateral edge is perpendicular to the base).
1) Lateral surface S side. = P main. ∙ N.
2) Full surface S full. = 2S main. + S side.
3) The volume of a straight parallelepiped V = S main. ∙ N.
Inclined parallelepiped.
At the base there is a parallelogram or rectangle or a rhombus or square, and the side edges are NOT perpendicular to the plane of the base.
1) Volume V = S main. ∙ H;
2) Volume V = S section. ∙ l, where l— lateral rib, S section - cross-sectional area of an inclined parallelepiped drawn perpendicular to the side edge l.
Straight prism.
Lateral surface S side. = P main. ∙ H;
Full surface S full = 2S main. + S side. ;
The volume of the straight prism V = S main. ∙ N.
Inclined prism.
The lateral and total surfaces, as well as the volume, can be found using the same formulas as in the case of a straight prism. If the cross-sectional area of the prism is known perpendicular to its lateral edge, then the volume V = S cross-section. ∙ l, where
l- lateral rib, S section - the area of the section perpendicular to the lateral rib l.
Pyramid.
1)
lateral surface S side. equal to the sum of the areas of the side faces of the pyramid;
2) full surface S full. = S main. + S side. ;
3) volume V = (1/3) S main. ∙ N.
4) A regular pyramid has a regular polygon at its base, and the top of the pyramid is projected to the center of this polygon, i.e., to the center of the circumscribed and inscribed circles.
5) Apothem l Is the height of the side face of the regular pyramid. The lateral surface of the regular pyramid is S side. = (½) P main. ∙ l.
Three perpendicular theorem.
A straight line drawn on a plane through the base of the inclined, perpendicular to its projection, and perpendicular to the most inclined.
Converse theorem. If a straight line on a plane is perpendicular to an inclined one, then it is perpendicular to the projection of this inclined one.
Truncated pyramid.
If S and s are respectively the areas of the bases of the truncated pyramid, then the volume of any truncated pyramid
where h is the height of the truncated pyramid.
Lateral surface of a regular truncated pyramid
where P and p, respectively, are the perimeters of the bases of a regular truncated pyramid,
l-apothem (the height of the side face of a regular truncated pyramid).
Cylinder.
Lateral surface S side. = 2πRH;
Full surface S full = 2πRH + 2πR 2 or S full. = 2πR (H + R);
Cylinder volume V = πR 2 H.
Cone.
Lateral surface S side. = πR l;
Full surface S full. = πR l+ πR 2 or S full. = πR ( l+ R);
The volume of the pyramid is V = (1/3) πR 2 H. Here l- generatrix, R - base radius, H - height.
Ball and sphere.
The area of the sphere S = 4πR 2; Ball volume V = (4/3) πR 3.
R is the radius of the sphere (ball).
Theorems and general information
I. Geometry
II. Planimetry without formulas.
The two corners are called adjacent, if they have one side in common, and the other two sides of these corners are additional half-lines.
1. The sum of adjacent angles is 180 ° .
The two corners are called vertical if the sides of one corner are complementary half-lines of the sides of the other.
2. The vertical angles are equal.
Angle equal to 90 ° is called right angle... Lines intersecting at right angles are called perpendicular.
3. Through each point of a straight line, you can draw and, moreover, only one perpendicular straight line.
Angle less than 90 ° is called sharp... Angle greater than 90 ° is called stupid.
4. Signs of equality of triangles.
- on both sides and the corner between them;
- along the side and two corners adjacent to it;
- on three sides.
The triangle is called isosceles if its two sides are equal.
Median triangle is called the segment connecting the top of the triangle with the middle of the opposite side.
Bisector a triangle is called a segment of a straight line enclosed between the vertex and the point of its intersection with the opposite side, which divides the angle in half.
Height a triangle is a segment of the perpendicular dropped from the apex of the triangle to the opposite side, or to its continuation.
The triangle is called rectangular if it has a right angle. In a right-angled triangle, the side opposite to the right angle is called hypotenuse... The other two sides are called legs.
5. Properties of the sides and corners of a right-angled triangle:
- the angles opposite to the legs are sharp;
- the hypotenuse is larger than any of the legs;
- the sum of the legs is greater than the hypotenuse.
6. Signs of equality of right-angled triangles:
- along the leg and sharp corner;
- on two legs;
- on the hypotenuse and leg;
- by hypotenuse and acute angle.
7. Isosceles triangle properties:
- in an isosceles triangle, the angles at the base are equal;
- if two angles in a triangle are equal, then it is isosceles;
In an isosceles triangle, the median drawn to the base is the bisector and the height;
- if in a triangle the median and bisector (or height and bisector, or median and height) drawn from any vertex coincide, then such a triangle is isosceles.
8. In a triangle opposite the larger side lies a larger angle, opposite the larger angle lies the larger side.
9. (Triangle inequality). Each triangle has a sum of two sides greater than the third side.
Outside corner triangle ABC at the vertex A is called the angle adjacent to the angle of the triangle at the vertex A.
10. The sum of the interior angles of the triangle:
The sum of any two angles of a triangle is less than 180 ° ;
Each triangle has two sharp corners;
The outer corner of a triangle is greater than any inner corner not adjacent to it;
The angles of a triangle add up to 180 ° ;
The outer angle of a triangle is equal to the sum of two other angles that are not adjacent to it.
The sum of the acute angles of a right triangle is 90 ° .
The segment connecting the midpoints of the sides of the triangle is called the middle line of the triangle.
11. The middle line of a triangle has the property - it is parallel to the base of the triangle and is equal to its half.
12. The length of the polyline is not less than the length of the line segment connecting its ends.
13. Properties of the midpoint perpendicular of the segment:
A point lying on the middle perpendicular is equally distant from the ends of the segment;
Any point equally distant from the ends of the line segment lies on the middle perpendicular.
14. Properties of the angle bisector:
Any point lying on the bisector of an angle is equally distant from the sides of the angle;
Any point equally distant from the sides of an angle lies on the bisector of the angle.
15. Existence of a circle circumscribed about a triangle:
All three perpendiculars of the triangle intersect at one point and this point is the center of the circumscribed circle. A circle circumscribed about a triangle always exists and it is unique;
The center of the circumscribed circle of a right-angled triangle is the midpoint of the hypotenuse.
16. Existence of a circle inscribed in a triangle:
All three bisectors of the triangle intersect at one point and this point is the center of the inscribed circle. A circle inscribed in a triangle always exists and it is unique.
17. Signs of parallelism of straight lines. Theorems on parallelism and perpendicularity of straight lines:
Two straight lines, parallel to the third, are parallel;
If at the intersection of two straight lines the third, the inner (outer) criss-crossing angles are equal, or the inner (outer) one-sided angles total 180 ° , then these lines are parallel;
If parallel straight lines are intersected by a third straight line, then the inner and outer angles lying crosswise are equal, and the inner and external unilateral the total angles are 180 ° ;
Two straight lines perpendicular to the same straight line are parallel;
A straight line perpendicular to one of two parallel straight lines is perpendicular to the other.
Circle- the set of all points of the plane equidistant from one point.
Chord- a segment connecting two points of a circle.
Diameter- a chord passing through the center.
Tangent- a straight line that has one common point with a circle.
Central corner- the angle with the apex in the center of the circle.
Inscribed corner- the angle with the apex on the circle, the sides of which intersect the circle.
18. Theorems related to the circle:
The radius drawn to the tangent point is perpendicular to the tangent;
The diameter passing through the middle of the chord is perpendicular to it;
The squared length of the tangent is equal to the product of the secant length by its outer part;
The central angle is measured by the degree measure of the arc on which it rests;
The inscribed angle is measured by half of the arc on which it rests, or complements half of it to 180 ° ;
Tangents drawn to a circle from one point are equal;
The product of the secant to its outer part is a constant value;
Parallelogram is called a quadrilateral whose opposite sides are pairwise parallel.
19. Signs of a parallelogram. Parallelogram properties:
Opposite sides are equal;
Opposite angles are equal;
The parallelogram diagonals are halved by the intersection point;
The sum of the squares of the diagonals is equal to the sum of the squares of all its sides;
If in a convex quadrangle the opposite sides are equal, then such a quadrangle is a parallelogram;
If in a convex quadrangle the opposite angles are equal, then such a quadrangle is a parallelogram;
If in a convex quadrangle the diagonals are halved by the intersection point, then such a quadrangle is a parallelogram;
The midpoints of the sides of any quadrangle are the vertices of the parallelogram.
A parallelogram, all sides of which are equal, is called rhombus.
20. Additional properties and features of a rhombus:
The diagonals of the rhombus are mutually perpendicular;
The diagonals of a rhombus are the bisectors of its interior angles;
If the diagonals of a parallelogram are mutually perpendicular, or are bisectors of the corresponding angles, then this parallelogram is a rhombus.
A parallelogram, all angles of which are straight, is called rectangle.
21. Additional properties and attributes of a rectangle:
The diagonals of the rectangle are equal;
If the diagonals of a parallelogram are equal, then such a parallelogram is a rectangle;
The midpoints of the sides of the rectangle are the vertices of the rhombus;
The midpoints of the sides of the rhombus are the vertices of the rectangle.
A rectangle with all sides equal is called square.
22. Additional properties and signs of a square:
The diagonals of the square are equal and perpendicular;
If the diagonals of a quadrilateral are equal and perpendicular, then such a quadrilateral is a square.
A quadrangle, two sides of which are parallel, is called trapezoid.
The segment connecting the midpoints of the lateral sides of the trapezoid is called the middle line of the trapezoid.
23. Trapezium properties:
- in an isosceles trapezoid, the angles at the base are equal;
- the segment connecting the midpoints of the diagonals of the trapezoid is equal to the half-difference of the bases of the trapezoid.
24. The middle line of a trapezoid has the property - it is parallel to the bases of the trapezoid and is equal to their half-sum.
25. Signs likeness triangles:
Two corners;
On two proportional sides and the angle between them;
On three proportional sides.
26. Signs of the similarity of right-angled triangles:
On a sharp corner;
By proportional legs;
By proportional leg and hypotenuse.
27. Ratios in polygons:
All regular polygons are similar to each other;
The sum of the angles of any convex polygon is 180 ° (n-2);
The sum of the outer angles of any convex polygon, taken one at each vertex, is 360 ° .
The perimeters of such polygons are referred to as their similar hand, and this ratio is equal to the coefficient of similarity;
The areas of similar polygons are referred to as the squares of their similar sides, and this ratio is equal to the square of the similarity coefficient;
The most important planimetry theorems:
28. Thales' theorem. If parallel straight lines intersecting the sides of the corner cut off equal segments on one side, then these straight lines also cut off equal segments on the other side.
29. Pythagorean theorem. In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs:.
30. The cosine theorem. In any triangle, the square of the side is equal to the sum of the squares of the other two sides, without their doubled product by the cosine of the angle between them:.
31. The theorem of sines. The sides of the triangle are proportional to the sines of the opposite angles: 
, where is the radius of the circle circumscribed about this triangle.
32. Three medians of a triangle intersect at one point, which divides each median in a ratio of 2: 1, counting from the apex of the triangle.
33. Three lines containing the heights of the triangle intersect at one point.
34. The area of a parallelogram is equal to the product of one of its sides by the height lowered to this side (or the product of the sides by the sine of the angle between them).
35. The area of a triangle is equal to half of the product of the side and the height, lowered to this side (or half of the product of the sides and the sine of the angle between them).
36. The area of the trapezoid is equal to the product of the half-sum of the bases and the height.
37. The area of the rhombus is half the product of the diagonals.
38. The area of any quadrangle is equal to half the product of its diagonals by the sine of the angle between them.
39. The bisector divides the side of a triangle into segments proportional to its other two sides.
40. In a right-angled triangle, the median drawn to the hypotenuse divides the triangle into two equal triangles.
41. The area of an isosceles trapezoid, the diagonals of which are mutually perpendicular, is equal to the square of its height:.
42. The sum of the opposite angles of a quadrilateral inscribed in a circle is 180 ° .
43. A quadrilateral can be described around a circle if the sums of the lengths of the opposite sides are equal.
III.Basic formulas for planimetry.
1. Arbitrary triangle.- from the side; - opposing angles; - semi-perimeter; - the radius of the circumscribed circle; - radius of the inscribed circle; - square; - height drawn to the side:
|
|
|||
Solving oblique triangles:
Cosine theorem:.
Sine theorem: 
.
The length of the median of a triangle is expressed by the formula:
.
The length of the side of the triangle through the medians is expressed by the formula:
.
The length of the bisector of a triangle is expressed by the formula:
,
Right triangle.- to athets; - hypotenuse; - projection of the legs to the hypotenuse:
|
|
Pythagorean theorem: .
Solving Right Triangles:
2. Equilateral triangle:
3. Arbitrary convex quadrilateral: - diagonals; - the angle between them; - square.
4. Parallelogram: - adjacent sides; - the angle between them; - the height drawn to the side; - square.
5.
Rhombus: 
6. Rectangle:
7. Square:
8. Trapezium:- bases; - height or distance between them; - the middle line of the trapezoid.
.
9. Described polygon(- semi-perimeter; - inscribed circle radius):
10. Regular polygon(- side of the correct - square; - the radius of the circumscribed circle; - radius of the inscribed circle):
11. Circle, circle(- radius; - circumference; - area of a circle):
12. Sector(- the length of the arc bounding the sector; - the degree measure of the central angle; - the radian measure of the central angle):
|
|
|
Objective 1.Area of a triangle ABC is 30 cm 2. On the side AC taken point D so that AD: DC = 2: 3. Perpendicular lengthDE held on the side of BC, is equal to 9 cm. Find BC.
Solution. Let's spend BD (see fig. 1.); triangles ABD and BDC have a total height Bf ; therefore, their areas are referred to as the lengths of the bases, i.e .:
AD: DC=2:3,
where 
18 cm 2.
On the other side 
, or, whence BC = 4 cm. Answer: BC = 4 cm.
Objective 2.In an isosceles triangle, the heights drawn to the base and to the side are 10 and 12 cm, respectively. Find the length of the base.
Solution. V ABC we have AB=
BC,
BD^
AC,
AE^
DC,
BD= 10 cm and AE= 12 cm (see Fig. 2). Let Rectangular TrianglesAEC
and BDC similar (angle Cgeneral); therefore, or 10: 12 = 5: 6. Applying the Pythagorean theorem to BDC, we have, i.e. ...
Transcript
1 Basic definitions, theorems and formulas for planimetry. Legend: ABC triangle with vertices A, B, С. circumscribed and inscribed circles. S is the area of the figure, d 1, d 2 are the diagonals of the quadrangle, the angle between straight lines a and b; signs, parallelism, perpendicularity, similarity, respectively. On definition, T theorem. T 1. (Signs of parallelism of straight lines, Fig. (6). О-1. А 1 В 1 С 1 ", ~ ABC (k is the coefficient of similarity), if their sides are proportional, and the corresponding angles are equal (Fig. 7): Two straight lines are parallel if: the inner cross-lying angles are equal:< 3 = < 5; внешние накрест лежащие УГЛЫ равны: < 1 = < 7; соответственные углы равны: <1 = < 5; сумма внутренних односторонних углов равна 180: < 2 + < 5= 180 ; сумма внешних односторонних углов равна 180: < 1 + < 6 = 180. Т 2 (признаки подобия). Два треугольника подобны, если: дня угла одного равны двум углам другого; дне стороны одного пропорциональны двум сторонам другого, а углы, заключенные между этими сторонами, равны; три стороны одного пропорциональны трем сторонам другого.
2 T 3. In such triangles, all their linear elements (with the same k) are proportional: sides, medians, bisectors, heights, radii of inscribed and circumscribed circles, etc. T 4 (Thales). Parallel straight lines intersecting the sides of the angle cut off proportional segments from them (Fig. 8): T 5. The sum of the angles of the triangle is 180. T 6. Three medians of the triangle intersect at one point, which divides each median into parts in a ratio of 2: 1, counting from the vertex (see Fig. 9): T 7. The middle line of the triangle, connecting the midpoints of the two sides, is parallel to the third side and equal to its half (Fig. 10): T 8. The bisector of the inner corner of the triangle divides the opposite side into parts proportional adjacent sides: BD: CD = AB: AC (see Fig. 11).
3 T 9. The inscribed angle (formed by two chords emanating from one point of the circle) is measured by half of the arc on which it rests (Fig. 12): T-10. The central angle formed by the two radii of the circle is measured by the arc on which it rests (see Fig. 12): T 11. The angle between the tangent and the chord drawn through the tangent point is measured by half of the arc between its sides (Fig. 13) : T 12. The angle between two secants with a vertex outside the circle is measured by the half difference of two arcs enclosed between its sides (Fig. 14): T 13. The tangents drawn to the circle from a common point outside the circle are equal: B A = BC. The angle between two tangents (the described angle) is measured by the half-difference of the larger and smaller arcs enclosed between the points of tangency (Fig. 15):
4 T 14. The angle between two chords with a top inside a circle is measured by the half-sum of two arcs, one of which is between its sides, the other between their extensions (Fig. 16): T 15. If two chords intersect inside circles, then the product of segments of one chord is equal to the product of other segments (see Fig. 16): AO OB = CO OD. T 16. If a tangent and a secant are drawn from a point outside the circle, then the square of the tangent is equal to the product of the secant segment by its outer part (Fig. 17): T 17. In a right-angled triangle (a, b - legs, c hypotenuse. H height, lowered to the hypotenuse, and c, bc, the projections of the legs to the hypotenuse) take place (Fig. 18): 1. Pythagorean formula: c 2 = a 2 + b 2 2. formulas 3. determination of trigonometric values (functions) of acute angles: 4. formulas for solving a right-angled triangle:
5 5. the center of a circle described about a right-angled triangle lies in the middle of the hypotenuse and T 18 (the theorem of sines). In an arbitrary triangle (Fig. 19) T-19 (cosine theorem). In an arbitrary triangle (Fig. 19): T 20. The sum of the squares of the lengths of the diagonals of the parallelogram is equal to the sum of the squares of the lengths of its sides: T 21. The center of a circle described in an angle lies on the bisector of this angle. The radius of the circle is perpendicular to the side of the corner and the tangency point. The center of a circle inscribed in a triangle is located at the intersection of the bisectors of the angles of the triangle. T 22. The center of the circle circumscribed about the triangle is located at the point of intersection of the mid-perpendiculars to the sides. T 23. In a quadrilateral circumscribed about a circle, the sums of opposite sides are equal. In particular, if an isosceles trapezoid is circumscribed about a circle, then its midline is equal to the lateral side. T 24. In a quadrangle inscribed in a circle, the sums of opposite angles are equal to 180. T 25. The area of a triangle is equal to
6 T 26. In a regular triangle with side a: T 27. In a regular n-gon (a n is the side of the n-gon, R is the radius of the circumscribed circle, r is the radius of the inscribed circle): T 28. The areas of similar triangles are referred to as squares of similar sides. O-2. Two figures are called equal if their areas are the same. T 29. The median divides the triangle into two equal parts. Three medians divide the triangle into six equal parts. The segments connecting the point of intersection of the medians with the vertices divide the triangle into three equal parts. T 30. In an arbitrary triangle, the median length is calculated as follows (Fig. 19): T 31. Formulas for the areas of quadrangles: a square with side a: S = a 2; rectangle with sides n. n li: S = a b; parallelogram with sides a and b: rhombus with side a and an acute angle between the sides: trapezoid with bases a and b:
7 convex quadrangle: T-32. Other formulas: area of a polygon circumscribed about a circle of radius r: S = p r; area of a circle of radius R: area of a solution sector (rad): circumference of a circle of radius R: length of an arc and or rad: All formulas for the surface area of volumetric bodies Total surface area of a cube a - side of a cube Formula for the surface area of a cube, (S):
8 Find the surface area of a rectangular parallelepiped a, b, c, - sides of a parallelepiped Formula for the surface area of a parallelepiped, (S): Calculation of the surface area of a cylinder r- base radius h- cylinder height π 3.14 Formula for the area of a lateral surface of a cylinder, (S side): Formula the area of the entire surface of the cylinder, (S): Find the surface area of the ball, formula R is the radius of the sphere π 3.14
9 Formula of the surface area of the ball (S): Surface area of the spherical sector R - radius of the sphere r - radius of the base of the cone = radius of the segment π 3.14 Formula of the surface area of the spherical sector, (S): Surface area of the spherical layer h - height of the spherical layer, segment KN R - the radius of the ball itself O - the center of the ball π 3.14 Formula of the lateral surface area of a spherical layer, (S):
10 Surface area of a spherical segment A spherical segment is the part of a sphere that is cut off by a plane. In this example, plane ABCD. R - the radius of the ball itself h - the height of the segment π 3.14 Formula of the surface area of the spherical segment, (S): The surface area of the regular pyramid through the apothem L - apothem (the lowered perpendicular OC from the vertex C, to the edge of the base AB) P- base perimeter S main - base area Formula for the lateral surface area of a regular pyramid (S side): Formula for the total surface area of a regular pyramid (S):
11 Lateral surface area of a regular truncated pyramid m - apothem of the pyramid, segment ok P - perimeter of the lower base, abcde p - perimeter of the upper base, abcde Formula for the lateral surface area of a regular truncated pyramid, (S): Surface area of a straight, circular cone R - radius of the base of the cone H - height L - generatrix of the cone π 3.14 Formula of the lateral surface area of the cone, through the radius (R) and the generatrix (L), (S side): Formula of the lateral surface area of the cone, through the radius (R) and height (H), (S side): The formula for the total surface area of the cone, through the radius (R) and the generatrix (L), (S):
12 Formula for the total surface area of a cone, through the radius (R) and height (H), (S): Formulas for the surface area of a truncated cone R - radius of the lower base r- radius of the upper base L - generatrix of the truncated cone π 3.14 Formula for the lateral surface area of a truncated cone , (S side): Formula for the total surface area of a truncated cone, (S): Calculating the volume of a cube All formulas for the volume of geometric bodies a - side of a cube Formula for the volume of a cube, (V):
13 Volume of a rectangular parallelepiped a, b, c- sides of a parallelepiped Formula of volume of a parallelepiped, (V): Formula for calculating the volume of a sphere R- radius of a sphere π 3.14 Volume of a sphere, (V): Volume of a spherical layer h- height of a spherical layer R- radius lower base r- radius of the upper base π 3.14
14 The volume of the spherical layer, (V): The volume of the spherical sector h - the height of the segment R - the radius of the sphere π 3.14 The volume of the spherical sector, (V): The volume of the spherical segment, formula The spherical segment is the part of the sphere that is cut off by the plane. In this example, plane ABCD. R - ball radius h - segment height π 3.14 Volume of a ball segment, (V):
15 How to calculate the volume of a cylinder? h- cylinder height r- base radius π 3.14 Cylinder volume, (V): How to find the volume of a cone? H- height of the cone R- radius of the base π 3.14 Volume of the cone, (V): Formula of the volume of the truncated cone R- radius of the lower base r- radius of the upper base h- height of the cone π 3.14
16 Volume of the truncated cone, (V): Calculation of the volume of the pyramid h - height of the pyramid S - area of the base ABCDE Volume of the pyramid, (V): Calculation of the volume of the truncated pyramid h - height of the pyramid S bottom - area of the lower base, ABCDE S top - area of the upper base , abcde Truncated pyramid volume, (V): Find the volume of the regular pyramid
17 A pyramid at the base, which has a regular polygon and faces equal triangles, is called regular. h - height of the pyramid a - side of the base of the pyramid n - number of sides of the polygon at the base Volume of a regular pyramid, (V): Volume of a regular triangular pyramid A pyramid with an equilateral triangle base and equal faces, isosceles triangles, is called a regular triangular pyramid. h - height of the pyramid a - side of the base Volume of a regular triangular pyramid, (V): Volume of a regular quadrangular pyramid A pyramid with a square base and equal faces, isosceles triangles, is called a regular quadrangular pyramid. h - height of the pyramid a - side of the base Volume of a regular quadrangular pyramid, (V):
18 Volume of a regular tetrahedron A regular tetrahedron is a pyramid in which all faces are equilateral triangles. a -edge of a tetrahedron Volume of a regular tetrahedron (V):
1. Definition: If two corners have a common side, and the other two sides are complementary rays, then these angles are called adjacent. Property: The sum of adjacent angles is 180 °. MOL + LON = 180 o 2. Property:
Arbitrary triangle In the formulas below, the following designations are used: a) from the length of the sides ABC lying opposite the angles A B and C, respectively b) the height of the median l l l bisector c) radius
Task 3, 6, 6. Planimetry Angular relations in plane figures Theorem. The sum of the adjacent angles is 80 0. and the adjacent angles Theorem. The bisectors of adjacent angles are mutually perpendicular. Theorem. Vertical
Task 6. Planimetry Angular relations in plane figures Theorem. Two straight lines parallel to the third are parallel. Theorem. If two parallel lines are intersected by a secant, then. Criss-cross corners
1. Areas of flat figures Area of a triangle: page 1 2. Midline 3. Triangles The sum of the angles of a triangle is 180. The obtuse angle between the bisectors of two angles of a triangle is 90 + half of the third
Quarter 1 1. The sum of the angles of a convex n-gon is equal to (n 2) 180. 2. A parallelogram is a quadrilateral whose opposite sides are pairwise parallel. 3. Properties of the parallelogram: 1)
Analysis of geometric statements 1. 1. Indicate the numbers of correct statements. 1) If two angles of one triangle are equal to two angles of another triangle, then such triangles are similar. 2) Vertical angles
1. See fig. 4. The angle between the intersecting chords is equal to the half-sum of the opposite arcs cut by the chords. 5. The angle between the two secants is equal to the half-difference of the arcs cut by the secants on the circle. 1 Questions
Gushchin D. D. REFERENCE MATERIALS FOR PREPARATION FOR THE USE ON MATH TASKS 3 AND 6: PLANIMETRY IT IS NECESSARY TO KNOW: TRIANGLES A triangle is a figure consisting of three points that do not lie on one straight line, and three
MUNICIPAL AUTONOMOUS EDUCATIONAL INSTITUTION GYMNASIUM 16 TYUMEN TOWN MAOU COLLEGE 16 TYUMEN TOWN Examination tickets in geometry under the basic general education program 8BZD class
Test 448 Vertical angles 1. If the angles are not vertical, then they are not equal. 2. Equal angles are vertical angles only if they are centrally symmetric. 3. If the angles are equal and their union has
1.2. Tests 31. The ratio of the lateral side to the diagonal of an isosceles trapezoid with bases 12 and 20, provided that the center of the circumscribed circle lies on a larger base, is 1) 1; 2) 0.5; 3) 0.8; 4)
1. Areas of flat figures Area of a triangle: page 1 2. Midline 3. Triangles The sum of the angles of a triangle is 180. The angles at the base of an isosceles triangle are equal. Bisector, median and
TASKS 20 OGE IN MATHEMATICS BEGINNING GEOMETRIC INFORMATION (INTERCEPTS, LINEARS AND ANGLES) 1) A point lying on the middle perpendicular to the segment is equidistant from the ends of this segment. 2) There are three
Circles Tangent and secant, mutual arrangement of circles A circle is the locus of points equidistant from one point, which is called the center of the circle Part of the plane lying
Test 250. Segment. Length The length of a line segment is 1 if it is: 1. the height of an equilateral triangle with side 2; 2.the third side of a triangle in which the other two sides are 1 and 2, and the angle
Analysis of geometric statements 1. Indicate the numbers of correct statements. 1) If two angles of one triangle are equal to two angles of another triangle, then such triangles are similar. 2) Vertical angles
The theoretical part of the exam for G-8 class. Know and understand (make a drawing and show in the figure) the following definitions and theorems (without proof) from the textbook G-8 A.G. Frozen Chapter 1 1. Sum of angles
The structure of the credit work in geometry grade 11/2013 / The work contains 10 tasks. Duration of work 120 minutes. Part 1. Tasks 1-7 tasks of the basic level of complexity (part B of the exam) with a short solution
MOU Lyceum at TPU GEOMETRY REFERENCE Planimetry Tomsk 003. TRIANGLES .. Right-angled triangle ... Metric ratios b legs with hypotenuse h height AH = c BH = .... Area b S =. b) +
Grade 11. Typical calculation on the topic "Round bodies". Option 1 1. The diagonal of the axial section of the cylinder is equal to a. Find the volume of a cylinder if it is known that its axial section is a square. 2. In a rectangular
20. Analysis of geometric statements Part 1. FIPI Task. List (circle) the numbers of true statements. I) Initial geometric information (segments, straight lines and angles) 1. Point lying on the median
Grade 11. Typical calculation on the topic "Round bodies". Option 16 1. The area of the base of the cylinder refers to the area of the axial section as π Find the angle between the diagonals of the axial section. 2. On the surface of the ball
The website of elementary mathematics Dmitry Gushchin www.mthnet.sp.ru Gushchin D. D. REFERENCE MATERIALS FOR PREPARING FOR THE USE IN MATH TASKS B3 AND B6: PLANIMETRY PROBLEMS Checked content elements and
V.A. Smirnov Open bank of tasks in geometry (planimetry) 2018-2019 academic year YEAR THEOREMS, PROPERTIES AND FORMULAS 1. The theorem on vertical angles. 2. The first sign of equality of triangles. 3. Second sign
1 Analysis of geometric statements Answers to the tasks are a word, phrase, number or sequence of words, numbers. Write down your answer without spaces, commas, or other additional characters.
Stereometry axioms 1. 2. 3. 4. 5. Consequences from the axioms 1. 2. Is the statement always true? 1. Any 3 points lie in the same plane. 1 2. Any 4 points lie in the same plane. 3. Any 3 points do not lie
REFERENCE MATERIALS FOR PREPARATION FOR THE USE ON MATHEMATICS Gushchin D. D. TASKS B3 AND B6: PLANIMETRY TASKS Verifiable content elements and activities: knowledge of the concepts of a triangle, a quadrangle,
60 2.2. Tests 161. If the sides of the base of a regular truncated pyramid are 6 and 4, and the dihedral angle at the base is 0, then the side surface of a regular triangular truncated pyramid is 1) 10; 2)
Examination material on geometry for 9th grade The tasks in the tickets are similar. Ticket 1 1. The first sign of equality of triangles. 2. Theorem on the relations between the sides and angles of a triangle.
В6 all tasks from the bank Using trigonometric functions. Right-angled triangle 27238. In triangle ABC, angle C is ,. Find AB. 27232. In triangle ABC, angle C is equal to ,. Find AC. 27235.
Assignments 1. Insert words (phrases) instead of gaps so that the statement is correct D-11. 1.1. A vector, the end of which coincides with a given point, and the origin with the origin, is called a given
Help B9 Polyhedra A polyhedron is a body whose surface consists of a finite number of flat polygons. Prism A prism is a polyhedron that consists of two flat polygons,
1 NUMBERS, FRACTIONS, MODULES Sets: Æ - empty set N = (1, 2, 3,) - set of natural numbers Z = - set of integers Q = - set of rational numbers (fractions) R set of real (real)
V.A. Smirnov, I.M. Smirnova GEOMETRY A guide for preparing for GIA Tasks for choosing correct statements 2015 1 INTRODUCTION This guide is intended to prepare for solving geometric problems of GIA in mathematics.
1. Which of these statements are correct? Write down their numbers. A circle can be described around any triangle. If in a parallelogram the diagonals are equal and perpendicular, then this parallelogram is a square.
Exam tickets in geometry 2017-18 academic year Ticket 1 1. Signs of equality of triangles. Signs of equality of right-angled triangles. 2. Bases BC and AD of trapezoid ABCD are equal respectively
Textbook on geometry Grade 10 Repetition of planimetry (tasks in pictures) For students of the Lyceum 1502 at MPEI I half-year Summary 1. Program of the colloquium on "Planimetry". 2. Contents
Mathematical dictations in geometry for VII and VIII grades (From work experience) VII grade Dictation 1 "The sum of the angles of a triangle" 1. Given a triangle MKL. Write down what is the sum of the angles of this triangle.
Tickets for the 8th grade geometry exam. Ticket 1. 1. Polygons 2. Meaning of Sin, Cos, tg (table) 1) If two angles of one triangle are equal to two angles of another triangle, then such triangles
Geometry short answer assignments Assignment. Solve the task. Give a short answer. 1. Find the distance from a point to the origin. 2. Find the distance from the point to the origin. 3. At what
Grade 7 1. Types of angles. An angle is called a right angle if it is equal to 90 0. An angle is called acute if it is less than 90 0. An angle is called obtuse if it is greater than 90 0, but less than 180 0. Right angle Acute angle Obtuse
Algebra Abbreviated multiplication formulas: Square of sum (+ = + + Square of difference (- = - + Difference of squares = (+ (Cube of sum (+ = + + + Cube of difference (- = - + - Sum of cubes) + = (+ (- + Difference of cubes
Grade 11. Typical calculation on the topic "Prism". Option 16 1. The base of the inclined prism is a rectangle with sides a and b. Two adjacent side faces make angles and with the base plane. Find volume
Three lateral ribs and inclined to the plane of the base at an angle α. The side of the base is α. Find the area of the resulting section. 17. In a regular quadrangular prism, the base area is 144 cm², and the height
Square L S = l =; a in Trapezoid O is the angle between the diagonals l is the middle line of the trapezoid Coordinate method l D) Let A (x; y), B (x; y), then the coordinates of the vector ABx x y) Let A (x; y), B (x; y), then
Ticket 1 1) Definition of a polygon. Vertices, sides, diagonals, and perimeter of a polygon. Formula for the sum of the angles of a convex polygon 2) Prove the theorem on the midline of a triangle. 3) Radius OB
Ticket 1 1. The first sign of equality of triangles. 2. Parallelogram. Definition, properties. 3. The task on the topic "Coordinates and vectors". Ticket 2 1. The second sign of equality of triangles. 2. Rectangle.
Geometry tickets for the transfer exam in grade 8 (textbook Geometry 7 9 by L. S. Atanasyan.) Each ticket contains 4 questions. In the first question, it is proposed to formulate and prove a theorem. In the second
Test 94. Isosceles triangle. Property In any isosceles triangle: 1. at least one median is its bisector; 2. at least one bisector is not its height; 3.at least two
Reference material on geometry. I. Parallel lines. Signs of parallelism of straight lines: 1. If at the intersection of two intersecting straight lines, the lying angles are equal, then the straight lines are parallel. 2. If when crossing
Inscribed and circumscribed circles A circle circumscribed about a triangle is a circle that passes through all its vertices. A single circle can be described around any triangle.
Task 6 Planimetry: tasks related to angles. Right-angled triangle: calculating angles 1. In a triangle, the angle is 90, sin A = 7 25. Find. 2. In a triangle, the angle is 90, sin A = 17 17. Find.
Pyramids. 11.1.5. The base of the quadrangular pyramid is a square. One of the side edges is perpendicular to the plane of the base, the other two are inclined to the base at an angle of 60. Find the full surface
Smirnov V.A., Smirnova I.M. GEOMETRY OF THE PROVISION PROBLEM 2015 Introduction This tutorial is intended for those who want to learn how to solve proof problems in geometry. It contains about four hundred
EXAMPLES OF PROBLEMS FOR PREPARATION FOR THE II-MU STAGE OF THE OLYMPIAD IN MATH PLANIMETRY OF A TRIANGLE 1. The length of one of the legs of a right triangle is 10 cm longer than the other, but less than the length of the hypotenuse
PROBLEM 9 RECTANGULAR TRIANGLE 1. In a triangle ABC, the angle C is equal to ,. Find AB. 2. In a triangle ABC, the angle C is ,. Find AB. 3. In a triangle ABC, the angle C is ,. Find AB. 4. In a triangle
Theme 1 REPEAT PLANIMETRY Practice 1 In class (5 numbers) 1. The bases of the trapezoid are equal to a and b (a> b). Find the length of the segment MN, the ends of which divide the sides AB and CD in relation AM: MB =
Prototypes of task 6 1. In triangle ABC, angle C is 90 0, AC = 4.8, 25. In triangle ABC, AC = BC, AB = 8, 33 tga. 7 4 33 sin A. Find AB. Find AC. 2. In triangle ABC, angle C is 90 0,
Geometry work for grade 8. 1. Type of work: intermediate certification in geometry in grade 8 Purpose of the work: assessment of the level of achievement by grade 8 students of the planned learning outcomes of geometry.
Requirements for the level of training of students As a result of studying the geometry course of the 8th grade, students should: know: - the definition of parallel straight lines, the formulation of the sign of parallel straight lines and consequences
Table of contents Abbreviated multiplication and factorization formulas ... Quadratic equation ... Parabola ... 3 Powers and roots ... 3 Logarithms ... 4 Progressions ... 4 Trigonometry ... 5 Trigonometric equations ...
Master class "Geometry and stereometry for the exam in mathematics, part 1. October 2017. To solve problems, you need knowledge about geometric shapes and their properties, calculating the areas of flat figures, volumes
Contents 1. Arithmetic progression 2. Arithmetic square root 3. Bisector 4. Inscribed circle 5. Convex quadrilateral 6. Geometric progression 7. Division with remainder 8. Divisibility
Grade 10. Typical calculation on the topic "Planimetry". Option 1 1. In an acute-angled triangle, the projections of two sides to the third are 4 and 2 cm. Find the projection of the medians on the same side. 2. In an isosceles triangle
1Dremova O.N. (, MBOU SOSH "Anninsky Lyceum")
1. Geometry grades 7-9: textbook. for general education. institutions / A.V. Pogorelov. - 10th ed. - M .: Education, 2016 .-- 240 p.
2.http: //ru.solverbook.com
3.http: //ege-study.ru
4.https: //reshyege.ru/
5.http: // www.fmclass.ru/math.phpid = 4850e0880794e
6.http: //tehtab.ru
7.https: //ege.sdamgia.ru/problemid = 50847
8.http: //alexlarin.net/ege17.html
This article is an abstract presentation of the main work. The full text of the scientific work, applications, illustrations and other additional materials are available on the website of the IV International Competition of Research and Creative Works of Students "Start in Science" at the link: https://school-science.ru/1017/7/770.
Hypothesis, relevance, goal, objectives of the project, object and subject of research, results
Target: Reveal, prove little-known theorems, properties of geometry.
Research objectives:
1. Study educational and reference literature.
2. Collect little-known theoretical material necessary for solving planimetric problems.
3. Understand the proofs of little-known theorems and properties.
4. Find and solve the problems of KIMs of the Unified State Exam, for the application of these little-known theorems and properties.
Relevance: In the exam, in mathematics tasks, there are often problems in geometry, the solution of which causes some difficulties and makes you spend a lot of time. The ability to solve such problems is an essential condition for successfully passing the USE at the profile level in mathematics. But there is a solution to this problem, some of these problems can be easily solved using theorems, properties that are little known, and they are not given attention in the school mathematics course. In my opinion, this can explain my interest in the research topic and its relevance.
Object of study: geometric problems of KIMs of the Unified State Exam.
Subject of study: little-known theorems and properties of planimetry.
Hypothesis: There are little-known theorems and properties of geometry, the knowledge of which will facilitate the solution of some planimetric problems of the KIMs of the Unified State Exam.
Research methods:
1) Theoretical analysis and search for information about little-known theorems and properties;
2) Proof of theorems and properties
3) Search and solution of problems using these theorems and properties
In mathematics, and in general in geometry, there is a huge number of different theorems and properties. Many theorems and properties are known for solving planimetric problems, which are relevant to this day, but are little known, and very useful for solving problems. When studying this subject, only the basic, well-known theorems and methods of solving geometric problems are mastered. But besides this, there are a fairly large number of different properties and theorems that simplify the solution of a particular problem, but few people know about them at all. In KIMs of the Unified State Exam, solving problems in geometry can be many times easier, knowing these little-known properties and theorems. In CMMs, geometry problems are found in numbers 8, 13, 15 and 16. Little-known theorems and properties described in my work simplify many times the solution of planimetric problems.
The bisector theorem for the angles of a triangle
Theorem: The bisector of the angle of a triangle divides the opposite side into segments proportional to the adjacent sides of the triangle.
Proof.
Consider the triangle ABC and the bisector of its angle B. Draw through the vertex C the line CM, parallel to the bisector BK, until the intersection at the point M by the continuation of the side AB. Since VK is the bisector of the angle ABC, then ∠ABK = ∠KVS. Further, ∠АВК = ∠ВМС, as the corresponding angles for parallel lines, and ∠КВС = ∠BCM, as criss-crossing angles for parallel lines. Hence, ∠BCM = ∠BMC, and therefore the triangle of the BMC is isosceles, whence BC = BM. By the theorem on parallel lines intersecting the sides of an angle, we have AK: KC = AB: BM = AB: BC, which was required to prove.
Let us consider the problems in the solution of which the property of the bisectors of a triangle is used.
Problem # 1. In triangle ABC, the bisector AH divides the side BC into segments whose lengths are 28 and 12. Find the perimeter of triangle ABC if AB - AC = 18.
ABC - triangle
AH - bisector
Let AC = X then AB = X + 18
By the property of the bisector of the angle alpha, AB · HC = BH · AC;
28 X = 12 (x + 18) x = 13.5,
means AC = 13.5, whence
AB = 13.5 + 18 = 31.5 BC = 28 + 12 = 40,
P = AB + BC + AC = 85
Median triangle theorem
Theorem. The medians of the triangle intersect at one point and are divided in it in a ratio of 2: 1, counting from the top.
Proof. Draw medians AA1 and CC1 in triangle A BC and denote their intersection point by M.
Through point C1 we draw a line parallel to AA1 and denote its intersection point with BC by D.
Then D is the midpoint of BA1, therefore, CA1: A1D = 2: 1.
By Thales' theorem, CM: MC1 = 2: 1. Thus, median AA1 intersects median CC1 at point M, dividing median CC1 in a 2: 1 ratio.
Similarly, the median BB1 intersects the median CC1 at the point dividing the median CC1 in a 2: 1 ratio, i.e. point M.
Problem number 1. Prove that the median of the triangle lies closer to the larger side, ie. if the triangle is ABC, AC> BC, then the median CC1 satisfies the inequality ACC1< BCC1.
Let's extend the median CC1 and set aside the segment C1B equal to AC1. Triangle AC1D is equal to triangle BC1C on two sides and the angle between them. Therefore, AD = BC, ADC1 = BCC1. In a triangle ACD AC> AD. Since there is a larger angle opposite the larger side of the triangle, then ADC1> ACD. Therefore, the inequality ACC1 Problem # 2. The area of triangle ABC is 1. Find the area of a triangle whose sides are equal to the medians of this triangle. ABC-triangle Let AA1, BB1, CC1 be the medians of triangle ABC intersecting at point M. Continue the median CC1 and set aside the segment C1D equal to MC1. The area of triangle BMC is 1/3, and its sides are equal to 2/3 of the medians of the original triangle. Therefore, the area of a triangle whose sides are equal to the medians of this triangle is 3/4. Let's derive a formula expressing the medians of a triangle through its sides. Let the sides of triangle ABC be equal to a, b, c. The required length of the median CD is denoted by mc. By the cosine theorem, we have: Adding these two equalities and taking into account that cosADC = -cosBDC, we obtain the equality: from which we find The middle line theorem of a triangle Theorem: the three middle lines of a triangle divide it into 4 equal triangles, similar to this one with a similarity coefficient of 1/2 Proof: Let ABC be a triangle. C1 - middle of AB, A1 - middle of BC, B1 - middle of AC. Let us prove that triangles AC1B1, BC1A1, A1B1C, C1B1A1 are equal. Since C1 A1 B1 is the middle, then AC1 = C1B, BА1 = A1C, AB1 = B1C. We use the middle line property: C1A1 = 1/2 AC = 1/2 (AB1 + B1C) = 1/2 (AB1 + AB1) = AB1 Similarly C1B1 = A1C, A1B1 = AC1. Then in triangles AC1B1, BА1С1, A1В1C, С1В1А1 AC1 = BC1 = A1B1 = A1B1 AB1 = C1A1 = B1C = C1A1 C1B1 = BA1 = A1C = C1B1 So the triangles are equal on three sides, it follows that A1 / B1 = A1C1 / AC = B1C1 / BC = ½ The theorem is proved. Let's consider the solution of problems using the property of the middle lines of a triangle. Problem 1. Given a triangle ABC with sides 9,4 and 7. Find the perimeter of triangle C1A1B1 whose vertices are the midpoints of these sides Given: triangle - ABC 9,4,7-sides of a triangle By the property of similarity of triangles: 3 middle lines of a triangle divide it into 4 equal triangles, similar to this one with a coefficient of 1/2. C1A1 = 9/2 = 4.5 A1B1 = 4/2 = 2 C1B1 = 7/2 = 3.5 hence the perimeter is = 4.5 + 2 + 3.5 = 10 Circle tangent property Theorem: the square of the tangent is equal to the product of the secant and its outer part. Proof. Draw segments AK and BK. Triangles AKM and BKM are similar since they have a common angle M. And the angles AKM and B are equal, since each of them is measured by half of the arc AK. Therefore MK / MA = MB / MK, or MK2 = MA MB. Examples of problem solving. Problem number 1. From point A outside the circle, a secant line 12 cm long and a tangent line are drawn, the length of which is 2 times less than the secant line that is inside the circle. find the length of the tangent. ACD secant If a tangent and a secant are drawn from one point to the circle, then the product of the entire secant by its outer part is equal to the square of the tangent, that is, AD · AC = AB2. Or AD (AD-2AB) = AB2. We substitute the known values: 12 (12-2АB) = АB2 or АB2 + 24 · АB-144. AB = -12 + 12v2 = 12 (v2-1) Property of the sides of the circumscribed quadrilateral Theorem: for a quadrilateral circumscribed about a circle, the sums of the lengths of opposite sides are equal Proof: By the tangent property AP = AQ, DP = DN, CN = CM, and BQ = BM, we obtain AB + CD = AQ + BQ + CN + DN and BC + + AD = BM + CM + AP + DP. Hence AB + CD = BC + AD Let's consider examples of problem solving. Problem number 1. The three sides of a quadrilateral circumscribed about a circle are related (in sequential order) as 1: 2: 3. Find the larger side of this quad if you know its perimeter to be 32. ABCD - quadrilateral AB: BC: CD = 1: 2: 3 Let side AB = x, then AD = 2x, and DC = 3x. By the property of the described quadrangle, the sums of the opposite sides are equal, and therefore x + 3x = BC + 2x, whence BC = 2x, then the perimeter of the quadrilateral is 8X. We get that x = 4, and the large side is 12. Problem number 2. A trapezoid is described around the circle, the perimeter of which is 40. Find its center line. ABCD-trapezoid, l - middle line Solution: The midline of the trapezoid is half the sum of the bases. Let the bases of the trapezoid be a and c, and the sides b and d. By the property of the described quadrilateral, a + c = b + d, and therefore the perimeter is 2 (a + c). We get that a + c = 20, whence L = 10 Peak Formula Pick's theorem: the area of a polygon is: where Г is the number of lattice nodes at the boundary of the polygon B is the number of lattice nodes inside the polygon. For example, to calculate the area of the quadrilateral shown in the figure, we consider: G = 7, B = 23, whence S = 7: 2 + 23 - 1 = 25.5. The area of any polygon drawn on checkered paper can be easily calculated by representing it as the sum or difference of the areas of right-angled triangles and rectangles, the sides of which follow the grid lines passing through the vertices of the drawn triangle. In some cases, you can even apply the ready-made formula for the area of a triangle or quadrangle. But in some cases, these methods are either impossible to apply, or the process of their application is time consuming and inconvenient. To calculate the area of the polygon shown in the figure, applying the Peak formula, we have: S = 8/2 + 19-1 = 22. Conclusion In the course of the research, the hypothesis was confirmed that in geometry there are theorems and properties little known from the school course that simplify the solution of some planimetric problems, including the tasks of the KIMs of the Unified State Exam. I managed to find such theorems and properties and apply them to solving problems, and prove that their application reduces huge solutions of some problems to solutions in a couple of minutes. The application of the theorems and properties described in my work in some cases allows you to solve the problem immediately and orally, and allows you to save more time on the exam and simply when solving them at school. I believe that the materials of my research can be useful for graduates in preparing for the exam in mathematics. Let's start with a few basic properties of different types of angles: Now let's move on to the properties of the triangle. Let there be an arbitrary triangle: Then, the sum of the angles of a triangle: Remember also that the sum of any two sides of a triangle is always greater than the third side... Area of a triangle through two sides and the angle between them: The area of the triangle through the side and the height dropped on it: The semi-perimeter of a triangle is found by the following formula: Heron's formula for the area of the triangle: Area of a triangle through the radius of the circumscribed circle: Formula for the median (median is a line drawn through some vertex and the middle of the opposite side in a triangle): Median properties: Bisector property (bisector is a line that divides some angle into two equal angles, i.e. in half): It's important to know: The center of the inscribed circle lies at the intersection of the bisectors(all three bisectors intersect at this one point). Bisector formulas: The main property of the heights of a triangle (height in a triangle is a line passing through some vertex of the triangle perpendicular to the opposite side): All three heights in the triangle intersect at one point. The position of the intersection point is determined by the type of triangle: Another useful property of triangle heights: Cosine theorem: Sine theorem: The center of the circle circumscribed about the triangle lies at the intersection of the median perpendiculars. All three mid-perpendiculars intersect at this one point. Mid perpendicular - a line drawn through the middle of the side of the triangle perpendicular to it. The radius of a circle inscribed in a regular triangle: The radius of a circle around a regular triangle: Regular triangle area: Pythagorean theorem for a right-angled triangle ( c- hypotenuse, a and b- legs): Radius of a circle inscribed in a right-angled triangle: Radius of a circle circumscribed around a right-angled triangle: Area of a right triangle ( h- the height lowered to the hypotenuse): Properties of the height dropped on the hypotenuse of a right triangle: Similar triangles- triangles in which the angles are respectively equal, and the sides of one are proportional to the similar sides of the other. In such triangles, the corresponding lines (heights, medians, bisectors, etc.) are proportional. Similar sides similar triangles - sides lying opposite equal angles. Similarity coefficient- number k equal to the ratio of the similar sides of similar triangles. The ratio of the perimeters of similar triangles is equal to the coefficient of similarity. The ratio of the lengths of the bisectors, medians, heights and mid-perpendiculars is equal to the coefficient of similarity. The ratio of the areas of similar triangles is equal to the square of the similarity coefficient. Signs of the similarity of triangles: Trapezoid- a quadrilateral with exactly one pair of opposite sides parallel. Length of the middle line of the trapezoid: Trapezium area: Some properties of trapezoids: Parallelogram- this is a quadrangle, in which the opposite sides are pairwise parallel, that is, they lie on parallel lines. The area of the parallelogram through the side and the height lowered onto it: The area of a parallelogram across two sides and the angle between them: Some properties of a parallelogram: Square- a quadrilateral in which all sides are equal, and all angles are equal to 90 degrees. The area of a square in terms of the length of its side: The area of a square in terms of the length of its diagonal: Square properties- these are all the properties of a parallelogram, rhombus and rectangle at the same time. Rhombus is a parallelogram with all sides equal. Area of a rhombus (the first formula is through two diagonals, the second is through the length of the side and the angle between the sides): Diamond properties: Rectangle is a parallelogram with all angles straight (equal to 90 degrees). Area of a rectangle across two adjacent sides: Rectangle properties: Area of an arbitrary convex quadrilateral through two diagonals and the angle between them: Relationship between the area of an arbitrary figure, its semi-perimeter and the radius of the inscribed circle(it is obvious that the formula is valid only for figures in which a circle can be inscribed, i.e., including for any triangles): Generalized Thales theorem: Parallel straight lines are cut off at secant proportional segments. Sum of angles n-gon: The central angle of the correct n-gon: The area of the correct n-gon: The theorem on proportional segments of chords: Tangent and secant theorem: Two secant theorem: Center and Inscribed Angle Theorem(the value of the central angle is twice the value of the inscribed angle if they rest on a common arc): Inscribed angles property (all inscribed angles based on a common arc are equal to each other): Center angles and chords property: Center corners and secant properties: Circumference: Circular arc length: Area of a circle: Sector area: Ring area: Circular segment area: Successful, diligent and responsible implementation of these three points will allow you to show excellent results at the CG, the maximum of what you are capable of. If you, as it seems to you, found an error in the training materials, please write about it by mail. You can also write about the error on the social network (). In the letter, indicate the subject (physics or mathematics), the title or number of the topic or test, the number of the problem, or the place in the text (page) where, in your opinion, there is an error. Also describe what the alleged error is. Your letter will not go unnoticed, the error will either be corrected, or you will be explained why it is not an error.
.
Bibliographic reference
Khvorov I.I. LITTLE KNOWN PLANIMETRY THEORES // International School Scientific Bulletin. - 2018. - No. 3-2. - S. 184-188;
URL: http://school-herald.ru/ru/article/view?id=544 (date accessed: 02/01/2020). 
Trapezoid
Parallelogram
Square
Rhombus and rectangle
Arbitrary shapes
Circle
Found a bug?
