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REFERENCE MATERIAL ON GEOMETRY FOR GRADES 7-11.
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Definition of a parallelogram.
A parallelogram is a quadrilateral whose opposite sides are parallel in pairs: AB||CD, AD||DC.
Opposite sides of a parallelogram are: AB=CD, AD=DC.
Opposite angles of a parallelogram are:
∠A=∠c,∠B=∠D.
The sum of the angles of a parallelogram adjacent to one side is 180°. For example, ∠ A+∠B=180°.
Any diagonal of a parallelogram divides it into two equal triangles. ∆ABD=∆BCD.
The diagonals of a parallelogram intersect and the intersection point is bisected. 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 the quadrilateral is a parallelogram.
- If opposite sides of a quadrilateral are pairwise equal, then the quadrilateral is a parallelogram.
- If the diagonals of a quadrilateral intersect and the intersection point is bisected, then the quadrilateral is a parallelogram.
The area of a parallelogram.
1) S=ah;
2) S=ab∙sinα;
A rectangle is a parallelogram with all right angles. ABCD- rectangle. A rectangle has all the properties of a parallelogram.
The diagonals of a rectangle are equal.
AC=BD. Let AC=d 1 and BD=d 2 , ∠COD=α.
d 1 \u003d 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 \u003d (d 2) 2 \u003d a 2 + b 2.
Rectangle area can be found using the formulas:
1) S=ab; 2) S=(½) d²∙sinα; (d is the diagonal of the rectangle).
Near any rectangle it is possible to describe a circle, 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 with all sides equal.
ABCD- rhombus.
A rhombus has all the properties of a parallelogram.
The diagonals of a rhombus are mutually perpendicular.
AC | B.D.
The diagonals of a rhombus are the bisectors of its angles.
Rhombus area.
1) S=ah;
2) S=a 2 ∙sinα;
3) S \u003d (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 a square are equal, the diagonals of a 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 \u003d (½) d 2.
Trapeze.
Bases of the trapezium AD||BC, MN-median line
Trapezium area is equal to the product of half the 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 sides are equal; base angles are equal.
The area of any quadrilateral.
- The area of any quadrangle is equal to half the product of its diagonals and the sine of the angle between them:
S=(½) d 1 ∙d 2 ∙sinβ.
- The area of any quadrilateral is equal to half the product of its perimeter and the radius of the inscribed circle:
Inscribed and circumscribed quadrilaterals.
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 quadrilateral are 180° each, then a circle can be circumscribed around a quadrilateral. The converse is also true.
If the sums of opposite sides of a quadrilateral are (a+c=b+d), then a circle can be inscribed in this quadrilateral. The converse is also true.
Circle, circle.
1) Circumference С=2πr;
2) The area of the circle S=πr 2 ;
3) Arc length AB:
4) AOB sector area:
5) Segment area (highlighted area):
("-" is taken if α<180°; «+» берут, если α>180°), ∠AOB=α – central angle. Arc l seen from the center O at an angle α.
In a right 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.
SΔ=(½) a∙b, where a and b are legs or SΔ\u003d (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 triangle.
Proportional segments in a right triangle.
The height drawn from the vertex of the right angle to the hypotenuse is the average proportional value between the projections of the legs on 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 are the middle members of the corresponding proportions).
Sine theorem.
In any triangle, the sides are proportional to the sines of the opposite angles.
Consequence from the sine theorem.
Each of the ratios of a side to the sine of the opposite angle is 2R, where R is the radius of the circle circumscribing the 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 doubling the product of these sides by the cosine of the angle between them.
Properties of an isosceles triangle.
In an isosceles triangle ( side lengths 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°.
External corner of a triangle(∠4) is equal to the sum of two inner ones not adjacent to it, i.e. ∠4=∠1+∠2.
Middle line of the triangle connects the midpoints of the sides of the triangle.
The middle line of the triangle is parallel to the base and equal to half of it: MN=AC/2.
Area of a triangle.
Heron formula.
The center of gravity of the triangle.
The center of gravity of a triangle is the point of intersection of the medians, which divides each median in a ratio of 2:1, counting from the top.
The length of the median drawn to side a:
The median divides the triangle into two triangles of equal area, the area of each of these two triangles is equal to half the area of the given triangle.
Angle bisector of a triangle.
1) The angle bisector 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:
3) All three bisectors of a triangle intersect at one point.
Center of a circle inscribed in a triangle, lies at the intersection of the angle bisectors of the triangle.
The area of the triangle S Δ =(½) P∙r, where P=a+b+c, r is the radius of the inscribed circle.
The radius of an inscribed circle can be found using the formula:
Center of a circle circumscribed about a triangle, lies at the intersection of the perpendicular bisectors of 
sides of the triangle.
The radius of a circle circumscribed about any triangle:
Radius of a circle inscribed about a right triangle, equal to half of the hypotenuse: R=AB/2;
The medians of right 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 around a regular n-gon.
Circle, inscribed into a regular n-gon.
sum of interior angles of any convex n-gon is 180°(n-2).
Sum of external angles of any convex0 n-gon is equal to 360°.
Rectangular parallelepiped.
All faces of a cuboid are rectangles. a, b, c - linear dimensions of a rectangular parallelepiped (length, width, height).
1) Diagonal of a rectangular parallelepiped d 2 \u003d a 2 + b 2 + c 2;
2) Side surface S side. =P main. ∙H or S side. =2 (a+b) c;
3) Complete surface S is complete. =2S main. +S side or
S full =2(ab+ac+bc);
4) Volume of a rectangular parallelepiped V=S main. ∙H or V=abc.
1) All faces of a cube are squares with side a.
2) Diagonal of the cube d=a√3.
3) Lateral surface of the cube S side. =4a 2 ;
4) The total surface of the cube S is complete. \u003d 6a 2;
5) Volume of the cube V=a 3 .
Right parallelepiped(at the base lies a parallelogram or a rhombus, the side edge is perpendicular to the base).
1) Side surface S side. =P main. ∙N.
2) The complete surface S is complete. =2S main. +S side
3) Volume of a straight parallelepiped V=S main. ∙N.
Inclined parallelepiped.
At the base of a parallelogram or a rectangle or a rhombus or a square, and the side edges are NOT perpendicular to the plane of the base.
1) Volume V=S main. ∙H;
2) Volume V=S sec. ∙ l, where l— side rib, S sec. - sectional area of an inclined parallelepiped drawn perpendicular to the side edge l.
direct prism.
Lateral surface S side. =P main. ∙H;
Total surface S total. =2S main. +S side ;
The volume of a direct prism V=S main. ∙N.
tilted 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 perpendicular to its side edge is known, then the volume V \u003d S sec. ∙ l, where
l- side rib, S sec. - cross-sectional area perpendicular to the lateral rib l.
Pyramid.
1)
side surface S side. equal to the sum of the areas of the side faces of the pyramid;
2) total surface S total. =S main. +S side ;
3) volume V=(1/3) S main. ∙N.
4) A regular pyramid has a regular polygon at the 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 a regular pyramid. Lateral surface of a regular pyramid S side. =(½) P main. ∙ l.
Theorem on three perpendiculars.
A straight line drawn in a plane through the base of an inclined one, perpendicular to its projection, is also perpendicular to the inclined itself.
Inverse theorem. If a straight line in a plane is perpendicular to an inclined one, then it is also perpendicular to the projection of this inclined one.
Truncated pyramid.
If S and s, respectively, are 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;
Total surface S total. =2πRH+2πR 2 or S full. =2πR(H+R);
Cylinder volume V=πR 2 H.
Cone.
Lateral surface S side. = πR l;
Total surface S total. =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 - radius of the base, H - height.
Ball and sphere.
The area of the sphere S=4πR 2 ; The volume of the ball is 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 angles are additional half-lines.
1. The sum of adjacent angles is 180 ° .
The two corners are called vertical if the sides of one angle are the complementary half-lines of the sides of the other.
2. Vertical angles are equal.
Angle equal to 90 ° , is called right angle. Lines that intersect at right angles are called perpendicular.
3. Through each point of a straight line it is possible to draw only one perpendicular to the straight line.
Angle less than 90 ° , is called sharp. Angle greater than 90 ° , is called stupid.
4. Signs of equality of triangles.
- on two sides and the angle between them;
- along the side and two corners adjacent to it;
- on three sides.
Triangle called isosceles if its two sides are equal.
median A triangle is a line segment that connects the vertex of the triangle with the midpoint of the opposite side.
bisector A triangle is called a straight line segment 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 a perpendicular dropped from the vertex 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 triangle, the side opposite the right angle is called hypotenuse. The other two sides are called legs.
5. Properties of sides and corners of a right triangle:
- the angles opposite the legs are acute;
- the hypotenuse is greater than any of the legs;
- the sum of the legs is greater than the hypotenuse.
6. Signs of equality of right triangles:
- along the leg and acute angle;
- on two legs;
- along the hypotenuse and leg;
- hypotenuse and acute angle.
7. Properties of an isosceles triangle:
- in an isosceles triangle, the angles at the base are equal;
- if in a triangle two angles are equal, then it is isosceles;
In an isosceles triangle, the median drawn to the base is the bisector and height;
- if in a triangle the median and the bisector (or the height and the bisector, or the median and the height) drawn from some vertex coincide, then such a triangle is isosceles.
8. In a triangle, a larger angle lies opposite the larger side, and a larger side lies opposite the larger angle.
9. (Triangle inequality). In every triangle, the sum of two sides is greater than the third side.
outside corner triangle ABC at vertex A is called the angle adjacent to the angle of the triangle at vertex A.
10. The sum of the interior angles of a triangle:
The sum of any two angles of a triangle is less than 180 ° ;
Each triangle has two acute angles;
The exterior angle of a triangle is greater than any interior angle not adjacent to it;
The sum of the angles of a triangle is 180 ° ;
An exterior angle of a triangle is equal to the sum of two other angles not adjacent to it.
The sum of the acute angles of a right triangle is 90 ° .
The line segment that joins the midpoints of the sides of a triangle is called midline of the triangle.
11. The middle line of a triangle has the property that it is parallel to the base of the triangle and equal to half of it.
12. The length of the broken line is not less than the length of the segment connecting its ends.
13. Properties of the perpendicular bisector of a segment:
A point lying on the perpendicular bisector is equidistant from the ends of the segment;
Any point equidistant from the ends of a segment lies on the perpendicular bisector.
14. Properties of the angle bisector:
Any point lying on the bisector of an angle is equidistant from the sides of the angle;
Any point equidistant from the sides of an angle lies on the bisector of the angle.
15. Existence of a circle circumscribed about a triangle:
All three perpendicular bisectors of a triangle intersect at one point, and this point is the center of the circumscribed circle. The circle circumscribed about a triangle always exists and is unique;
The center of the circumscribed circle of a right triangle is the midpoint of the hypotenuse.
16. Existence of a circle inscribed in a triangle:
All three bisectors of a triangle intersect at one point, and this point is the center of the inscribed circle. A circle inscribed in a triangle always exists and is unique.
17. Signs of parallel lines. Theorems on parallelism and perpendicularity of lines:
Two lines parallel to a third are parallel;
If at the intersection of two straight lines by a third, the internal (external) crosswise lying angles are equal, or the internal (external) unilateral angles sum to 180 ° , then these lines are parallel;
If parallel lines are intersected by a third line, then the interior and exterior crosswise angles are equal, and the interior and external unilateral angles add up to 180 ° ;
Two lines perpendicular to the same line are parallel;
A line perpendicular to one of two parallel lines is also perpendicular to the other.
Circle is the set of all points in the plane equidistant from one point.
Chord A line segment that connects two points on a circle.
Diameter is a chord passing through the center.
Tangent A straight line that has one common point with a circle.
Central corner is the angle with the vertex at the center of the circle.
Inscribed angle An angle with a vertex on a circle whose sides intersect the circle.
18. Theorems related to the circle:
The radius drawn to the point of contact is perpendicular to the tangent;
The diameter passing through the middle of the chord is perpendicular to it;
The square of the length of the tangent is equal to the product of the length of the secant and its outer part;
The central angle is measured by the degree measure of the arc on which it rests;
An inscribed angle is measured by half the arc on which it rests, or complements half of it up to 180 ° ;
Tangents drawn to a circle from one point are equal;
The product of a secant by 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 diagonals of a parallelogram are bisected by the point of intersection;
The sum of the squares of the diagonals is equal to the sum of the squares of all its sides;
If in a convex quadrilateral the opposite sides are equal, then such a quadrilateral is a parallelogram;
If opposite angles in a convex quadrilateral are equal, then such a quadrilateral is a parallelogram;
If in a convex quadrilateral the diagonals are bisected by the point of intersection, then such a quadrilateral is a parallelogram;
The midpoints of the sides of any quadrilateral are the vertices of the parallelogram.
A parallelogram with all sides equal is called rhombus.
20. Additional properties and signs 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 of whose angles are right angles is called rectangle.
21. Additional properties and features of a rectangle:
The diagonals of a 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 a square are equal and perpendicular;
If the diagonals of a quadrilateral are equal and perpendicular, then the quadrilateral is a square.
A quadrilateral whose two sides are parallel is called trapezoid.
The line segment that joins the midpoints of the sides of a trapezoid is called midline of the trapezoid.
23. Trapeze 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 midline of a trapezoid has the property that it is parallel to the bases of the trapezoid and equal to their half-sum.
25. Signs similarities triangles:
At two corners;
On two proportional sides and the angle between them;
on three proportional sides.
26. Signs of similarity of right triangles:
On a sharp corner;
By proportional legs;
By proportional leg and hypotenuse.
27. Relationships 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 exterior angles of any convex polygon, taken one at each vertex, is 360 ° .
The perimeters of similar polygons are related as they are similar sides, and this ratio is equal to the coefficient of similarity;
The areas of similar polygons are related as the squares of their similar sides, and this ratio is equal to the square of the similarity coefficient;
The most important theorems of planimetry:
28. Thales' theorem. If parallel lines intersecting the sides of an angle cut off equal segments on one side, then these lines also cut off equal segments on the other side.
29. The Pythagorean theorem. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs: .
30. Cosine theorem. In any triangle, the square of a side is equal to the sum of the squares of the other two sides without twice their product times the cosine of the angle between them: .
31. Sine theorem. The sides of a 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 top of the triangle.
33. Three lines containing the heights of a 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 the product of the side and the height lowered to this side (or half the product of the sides and the sine of the angle between them).
36. The area of a trapezoid is equal to the product of half the sum of the bases and the height.
37. The area of a rhombus is half the product of the diagonals.
38. The area of any quadrilateral is equal to half the product of its diagonals and the sine of the angle between them.
39. The bisector divides the side of the triangle into segments proportional to its other two sides.
40. In a right triangle, the median drawn to the hypotenuse divides the triangle into two triangles of equal area.
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 opposite sides are equal.
III.Basic formulas of planimetry.
1. Arbitrary triangle.- from the side; - opposite angles - semiperimeter; - radius of the circumscribed circle; - radius of the inscribed circle; - area; - height drawn to the side:
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Solution of 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 athety; - hypotenuse; - projections of the legs on the hypotenuse:
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Pythagorean theorem: .
Solution of right triangles:
2. Equilateral triangle:
3. Arbitrary convex quadrilateral: - diagonal; - the angle between them; - area.
4. Parallelogram: - adjacent sides; - the angle between them; - height drawn to the side; - area.
5.
Rhombus: 
6. Rectangle:
7. Square:
8. Trapeze:- grounds; - height or distance between them; - middle line of the trapezoid.
.
9. Circumscribed polygon(- semiperimeter; - radius of the inscribed circle):
10. regular polygon(- side of the correct - square; - radius of the circumscribed circle; - radius of the inscribed circle):
11. Circle, circle(- radius; - circumference; - area of a circle):
12. Sector(- length of the arc bounding the sector; - degree measure of the central angle; - radian measure of the central angle):
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Task 1.Area of a triangle ABC is 30cm 2. on the side AC the point D is taken so that AD : DC =2:3. Perpendicular lengthDE held to BC side, is equal to 9 cm. Find BC.
Solution. Let's spend BD (see Fig.1.); triangles ABD and BDC have a common height bf ; therefore, their areas are related as the lengths of the bases, i.e.:
AD : DC=2:3,
where 
18 cm2.
On the other hand 
, or , whence BC \u003d 4 cm. Answer: BC \u003d 4 cm.
Task 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. IN ABC we have AB=
BC,
BD^
AC,
AE^
DC,
BD=10 cm and AE\u003d 12 cm (see Fig. 2). Let right 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 of planimetry. Notation: ABC triangle with vertices A, B, C. a = BC, b = AC, c = AB its sides, respectively, median, bisector, height drawn to side a, P - perimeter, semi-perimeter, R and r radii, respectively circumscribed and inscribed circles. S is the area of the figure, d 1,d 2 are the diagonals of the quadrilateral, the angle between the lines a and b; signs, parallelism, perpendicularity, similarity, respectively. O definition, T theorem. T 1. (Signs of parallel lines, Fig. (6). O-1. A 1 B 1 C 1 ", ~ ABC (k - similarity coefficient), if their sides are proportional, and the corresponding angles are equal (Fig. 7): Two lines are parallel if: the interior diagonally 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 similar 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. The three medians of the triangle intersect at one point, which divides each median into parts in a ratio of 2: 1, counting from the top (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 half of it (Fig. 10): T 8. The bisector of the inner angle of the triangle divides the opposite side into parts proportional to 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 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 point of contact is measured by half the arc enclosed 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 located outside the circle are equal: B A \u003d BC. The angle between two tangents (described angle) is measured by the half-difference of the larger and smaller arcs enclosed between the tangent points (Fig. 15):
4 T 14. The angle between two chords with a vertex inside the circle is measured by the half-sum of two arcs, one of which is enclosed between its sides, the other between their continuations (Fig. 16): T 15. If two chords intersect inside the circle, then the product of segments of one chord is equal to the product of segments of the other (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 triangle (a, b - legs, c hypotenuse. h height, lowered to the hypotenuse, and c, bc projections of the legs to the hypotenuse) take place (Fig. 18): 1. Pythagorean formula: c 2 \u003d a 2 + b 2 2. formulas 3. definition of trigonometric quantities (functions) of acute angles: 4. formulas for solving a right triangle:
5 5. The center of the circle circumscribed about a right-angled triangle lies in the middle of the hypotenuse and T 18 (sine theorem). 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 a 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 point of contact. The center of a circle inscribed in a triangle is at the intersection point of the bisectors of the angles of the triangle. T 22. The center of a circle circumscribed about a triangle is located at the point of intersection of the perpendicular bisectors 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 180. T 25. The area of a triangle is
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 related 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 length of the median is calculated as follows (Fig. 19): T 31. Formulas for the areas of quadrangles: a square with side a: S \u003d 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 quadrilateral: 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 sector of the solution (rad): circumference of a circle of radius R: arc length 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, - the sides of the parallelepiped The formula for the surface area of the parallelepiped, (S): Calculation of the surface area of the cylinder r- radius of the base h- height of the cylinder π 3.14 The formula for the area of the lateral surface of the cylinder, (S side): Formula the area of the entire surface of the cylinder, (S): Find the surface area of the ball, the formula R is the radius of the sphere π 3.14
9 Sphere surface area formula (S): Spherical sector surface area R - ball radius r - cone base radius = segment radius π 3.14 Spherical sector surface area formula, (S): Spherical layer surface area h - spherical layer height, segment KN R - the radius of the ball itself O - the center of the ball π 3.14 The formula for the area of the lateral surface of the spherical layer, (S):
10 Surface area of a spherical segment A spherical segment is a part of a sphere 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 The formula for the surface area of the spherical segment, (S): The surface area of \u200b\u200bthe regular pyramid through the apothem L - the apothem (the lowered perpendicular OC from the top C, to the edge of the base AB) P - the perimeter of the base S base - base area The formula for the area of the lateral surface of a regular pyramid (S side): The formula for the area of the full surface of a regular pyramid (S):
11 The area of the lateral surface of a regular truncated pyramid m - the apothem of the pyramid, segment ok P - the perimeter of the lower base, abcde p - the perimeter of the upper base, abcde H - height L - generatrix of the cone π 3.14 The formula for the area of the lateral surface of the cone, through the radius (R) and generatrix (L), (S side): The formula for the area of the lateral surface of the cone, through the radius (R) and height (H), (S side): The formula for the area of the full surface of the cone, through the radius (R) and generatrix (L), (S):
12 The formula for the area of the full surface of the cone, through the radius (R) and height (H), (S): Formulas for the surface area of the 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 area of the lateral surface of the truncated cone , (S side): Formula for the area of the total surface of a truncated cone, (S): Calculation of the volume of a cube All formulas for the volume of geometric bodies a - side of the cube Formula for the volume of a cube, (V):
13 Volume of a rectangular parallelepiped a, b, c-sides of a parallelepiped Formula for the volume of a parallelepiped, (V): Formula for calculating the volume of a ball R- radius of the ball π 3.14 Volume of the ball, (V): Volume of the spherical layer h- height of the spherical layer R- radius lower base r-radius of 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 ball 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 ball cut off by the plane. In this example, plane ABCD. R - ball radius h - segment height π 3.14 Ball segment volume, (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 for the volume of a 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 lower - area of the lower base, ABCDE S top - area of \u200b\u200bthe upper base , abcde Volume of a truncated pyramid, (V): Find the volume of a regular pyramid
17 A pyramid at the base, which is a regular polygon and faces equal triangles, is called regular. h - the height of the pyramid a - the side of the base of the pyramid n - the number of sides of the polygon at the base The volume of a regular pyramid, (V): The volume of a regular triangular pyramid A pyramid whose base is an equilateral triangle and the faces are equal, isosceles triangles, is called a regular triangular pyramid. h - the height of the pyramid a - side of the base Volume of a regular triangular pyramid, (V): Volume of a regular quadrangular pyramid A pyramid whose base is a square and faces are equal, isosceles triangles, is called a regular quadrangular pyramid. h - the height of the pyramid a - the side of the base The volume of a regular quadrangular pyramid, (V):
18 The volume of a regular tetrahedron A regular tetrahedron is a pyramid whose all faces are equilateral triangles. a - edge of the tetrahedron The volume of a regular tetrahedron (V):
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1Dryomova O.N. (, MBOU secondary school "Anninsky Lyceum")
1. Geometry grades 7-9: textbook. for general education institutions / A.V. Pogorelov. – 10th ed. – M.: Enlightenment, 2016. – 240 p.
2. http://en.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 a summary 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 for 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 tasks in mathematics, 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 the successful completion of the exam 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 USE.
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 KIMs of the Unified State Examination.
Research methods:
1) Theoretical analysis and search for information about little-known theorems and properties;
2) Proof of theorems and properties
3) Finding and solving problems using these theorems and properties
In mathematics, and in general in geometry, there is a huge number of different theorems, 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 for solving geometric problems are assimilated. 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 Examination, solving problems in geometry can be much easier, knowing these little-known properties and theorems. In CMMs, problems in geometry are found in numbers 8, 13, 15 and 16. Little-known theorems and properties described in my work simplify the solution of planimetric problems many times over.
Triangle angle bisector theorem
Theorem: The angle bisector 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. Let us draw a straight line SM through the vertex C, parallel to the bisector BK, until it intersects at the point M by the continuation of the side AB. Since VC is the bisector of the angle ABC, then ∠ABK = ∠KBC. Further, ∠ABK = ∠BMC, as the corresponding angles at parallel lines, and ∠KBC = ∠BCM, as the cross-lying angles at parallel lines. Hence ∠VCM = ∠VMS, and therefore the triangle VMS is isosceles, hence BC = VM. By the theorem on parallel lines intersecting the sides of an angle, we have AK: KS = AB: VM = AB: BC, which was to be proved.
Let's consider the problems in the solution of which the property of the bisectors of a triangle is used.
Problem number 1. In triangle ABC, the bisector AH divides 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 angle bisector alpha, AB·HC = BH·AC;
28 X \u003d 12 (x + 18) x \u003d 13.5,
so AC = 13.5, whence
AB = 13.5 + 18 = 31.5BC = 28 + 12 = 40,
P=AB+BC+AC=85
Triangle median theorem
Theorem. The medians of a triangle intersect at one point and divide at it in a ratio of 2:1, counting from the top.
Proof. In the triangle A BC we draw the medians AA1 and CC1 and denote their intersection point by M.
Through the point C1 we draw a line parallel to AA1 and denote its intersection point with BC as D.
Then D is the middle of BA1, hence CA1:A1D = 2:1.
By the Thales theorem, CM:MC1 = 2:1. Thus median AA1 intersects median CC1 at point M dividing median CC1 in a ratio of 2:1.
Similarly, median BB1 intersects median CC1 at a point dividing median CC1 in a ratio of 2:1, i.e. point M.
Problem 1. Prove that the median of the triangle lies closer to the larger side, i.e. if in the triangle ABC, AC>BC, then the median CC1 satisfies the inequality ACC1< BCC1.
We continue the median CC1 and set aside the segment C1B equal to AC1. Triangle AC1D is equal to triangle BC1C in two sides and the angle between them. Therefore, AD = BC, ADC1 = BCC1. In 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 number 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. We 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 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 in terms of its sides. Let the sides of triangle ABC be a, b, c. Denote the desired length of the median CD by mc. By the law of cosines we have: Adding these two equalities and taking into account that cosADC = -cosBDC, we obtain the equality: from which we find Triangle midline theorem Theorem: three middle lines of a triangle divide it into 4 equal triangles similar to the given one with a similarity coefficient of ½ Proof: Let ABC be a triangle. C1 is the middle of AB, A1 is the middle of BC, B1 is the middle of AC. Let us prove that triangles AC1B1, BC1A1, A1B1C, C1B1A1 are equal. Since C1 A1 B1 is the middle, then AC1 \u003d C1B, BA1 \u003d A1C, AB1 \u003d 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, BA1C1, A1B1C, C1B1A1 AC1 = BC1 = A1B1 = A1B1 AB1 = C1A1 = B1C = C1A1 C1B1 = BA1 = A1C = C1B1 So the triangles are congruent on three sides, it follows that A1/B1 = A1C1/AC = B1C1/BC = ½ The theorem has been proven. Consider solving problems using the property of the middle lines of a triangle. Task number 1. Given a triangle ABC with sides 9,4 and 7. Find the perimeter of the triangle C1A1B1 whose vertices are the midpoints of these sides Given: triangle - ABC 9,4,7-sides of a triangle According to the similarity property 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 Property of a tangent to a circle Theorem: the square of the tangent is equal to the product of the secant and its outer part. Proof. Let's draw segments AK and BK. Triangles AKM and BKM are similar because 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. Task No. 1. From point A outside the circle, a secant is drawn, 12 cm long and a tangent, the length of which is 2 times less than the segment of the secant located 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, i.e. AD·AC = AB2. Or AD (AD-2AB) = AB2. Substitute the known values: 12(12-2AB) = AB2 or AB2 + 24 AB-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 get that AB + CD = AQ + BQ + CN + DN and BC + + AD = BM + CM + AP + DP. Consequently AB+CD=BC+AD Consider examples of problem solving. Problem No. 1. Three sides of a quadrangle circumscribed about a circle are related (in sequential order) as 1:2:3. Find the longest side of this quadrilateral if its perimeter is known 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 circumscribed quadrilateral, the sums of opposite sides are equal, and hence x + 3x = BC + 2x, whence BC = 2x, then the perimeter of the quadrilateral is 8X. We get that x \u003d 4, and the largest side is 12. Problem number 2. A trapezoid is circumscribed around a circle, the perimeter of which is 40. Find its midline. ABCD-trapezium, l - middle line Solution: The midline of a 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 circumscribed quadrilateral, a + c = b + d, and hence 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 on 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: D = 7, V = 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 whose sides 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 \u200b\u200ba triangle or quadrilateral. But in some cases, these methods are either impossible to apply, or the process of their application is time-consuming, inconvenient. To calculate the area of the polygon shown in the figure, using the Peak formula, we have: S \u003d 8/2 + 19-1 \u003d 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, which simplify the solution of some planimetric problems, including the problems of KIMs of the Unified State Examination. I was able to find such theorems and properties and apply them to problem solving, and prove that applying them reduces huge solutions to 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 just when solving them at school. I believe that the materials of my research can be useful to graduates in preparing for the exam in mathematics. To begin with, we indicate several basic properties of various types of corners: Now let's move on to the properties of the triangle. Let there be an arbitrary triangle: Then, 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. The area of a triangle in terms of two sides and the angle between them: The area of a triangle given a side and the height subtracted from it: The semiperimeter of a triangle is found by the following formula: Heron's formula for the area of a triangle: The area of a triangle in terms of the radius of the circumscribed circle: Median formula (median is a line drawn through some vertex and the midpoint of the opposite side in a triangle): Median properties: Bisector property (a bisector is a line that divides some angle into two equal angles, i.e. in half): It's important to know: The center of a circle inscribed in a triangle 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 (the height in a triangle is a line passing through some vertex of the triangle perpendicular to the opposite side): All three heights in a 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 mid-perpendiculars. All three median perpendiculars intersect at this one point. The perpendicular midpoint is a line drawn through the midpoint of a side of a triangle perpendicular to it. Radius of a circle inscribed in a regular triangle: Radius of a circle circumscribed about a regular triangle: Area of a right triangle: Pythagorean theorem for a right triangle ( c- hypotenuse, a And b- legs): Radius of a circle inscribed in a right triangle: Radius of a circle circumscribed around a right triangle: Area of a right triangle ( h- height lowered to the hypotenuse): Properties of the height dropped to 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 similar triangles, the corresponding lines (heights, medians, bisectors, etc.) are proportional. Related parties similar triangles - sides lying opposite equal angles. similarity coefficient- number k equal to the ratio of 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 bisectors, medians, heights and mid-perpendiculars is equal to the similarity coefficient. The ratio of the areas of similar triangles is equal to the square of the similarity coefficient. Signs of similarity of triangles: Trapeze A quadrilateral with exactly one pair of opposite sides parallel. The length of the midline of the trapezium: Trapezium area: Some properties of trapezoids: Parallelogram is a quadrilateral whose opposite sides are pairwise parallel, that is, they lie on parallel lines. The area of a parallelogram through the side and the height lowered on it: The area of a parallelogram through two sides and the angle between them: Some properties of a parallelogram: Square A quadrilateral with all sides equal and all angles 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. The 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): Rhombus properties: Rectangle is a parallelogram in which all angles are right (equal to 90 degrees). Area of a rectangle in terms of 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 semiperimeter and the radius of the inscribed circle(obviously, the formula is valid only for figures in which a circle can be inscribed, i.e. including for any triangles): Generalized Thales theorem: Parallel lines cut proportional segments at secants. Sum of angles n-square: Central angle of the correct n-square: Correct area n-square: Theorem on proportional segments of chords: Tangent and secant theorem: Two secant theorem: Central and inscribed angle theorem(the value of the central angle is twice the value of the inscribed angle if they are based on a common arc): Property of inscribed angles (all inscribed angles based on a common arc are equal to each other): Property of central angles and chords: Property of central angles and secants: Circumference: Arc length: Area of a circle: Sector area: Ring area: Area of a circular segment: Successful, diligent and responsible implementation of these three points will allow you to show an excellent result on the CT, the maximum of what you are capable of. If you, as it seems to you, found an error in the training materials, then 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 name or number of the topic or test, the number of the task, 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 a mistake.
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Bibliographic link
Khvorov I.I. LITTLE KNOWN THEOREMS OF PLANIMETRY // International School Scientific Bulletin. - 2018. - No. 3-2. – P. 184-188;
URL: http://school-herald.ru/ru/article/view?id=544 (date of access: 01/02/2020). 
Trapeze
Parallelogram
Square
Rhombus and Rectangle
Arbitrary figures
Circle
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