1 cknowledgement: Scott, Foresman. Geometry. SIMILR TRINGLS 1. efinition: ratio represents the comparison of two quantities. In figure, ratio of blue squares to white squares is 3 : 5 2. efinition: proportion is an equation which states that two ratios are equal. 2 3 = 4 6 The numbers 2 and 3 are proportional to 4 and 6. 3. efinition: Two convex polygons are similar if and only if there is a correspondence between their vertices such that corresponding angles are equal and corresponding sides are proportional.
2 4. Theorem: The ratio of the perimeter of two similar convex polygons is equal to the ratio of lengths of any two corresponding sides. 5. xiom: If two angles of one triangle are equal to two angles of another triangle, the triangles are similar. F 6. Theorem: Similarity of triangles is reflexive, symmetric and transitive. 7. Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides them proportionally. =
3 8. Theorem: If a line not containing a vertex of a triangle cuts off on two of its sides segments whose lengths are proportional to the lengths of these sides, then this line is parallel to the third side of the triangle. = 9. Theorem: If three parallel lines intersect two transversals, then the parallel lines divide transversals proportionally. F F = F
4 10. Theorem: If an angle of one triangle is equal to an angle of a second triangle, and if the lengths of the sides including these angles are proportional, then the triangles are similar. SS F =, = F F 11. Theorem: If the lengths of the sides of one triangle are proportional to the lengths of the sides of a second triangle, then the triangles are similar. SSS F = F = F F
5 12. Theorem: In similar triangles, the lengths of bisectors of corresponding angles are proportional to the lengths of corresponding sides. 3 4 1 2 X 1 = 2 = X Y Y 3 = 4 F 13. Theorem: In similar triangles, the lengths of altitudes from corresponding vertices are proportional to the lengths of corresponding sides. = X Y X X Y Y F F
6 14. Theorem: In similar triangles, the lengths of medians from corresponding vertices are proportional to the lengths of corresponding sides. X X = X and Y = YF = X Y Y F 15. Theorem: The bisector of an angle of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of the two sides adjacent to the segments. OR The bisector of an angle of a triangle divides the opposite side in the ratio
7 of the sides containing the angle. 1 2 1 = 2 = 16. Theorem: If an altitude is drawn to the hypotenuse of a right triangle, then the new triangles formed are similar to the given triangle and to each other. 17. efinition: The geometric mean of two positive numbers a and b is the a x positive number x such that. x b xample: Suppose an investment of X earns 25% in the first year and 80% in the
8 second year, then the average annual rate of return is 1.25 1. 80 = 1.5 (geometric mean of the rates of two years). Reference: When Less is More, M 18. Theorem: The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments into which the altitude separates the hypotenuse. h a b c 2 = ab a h a h = h b h b
9 19. Theorem: If the altitude to the hypotenuse is drawn in a right triangle, then the length of either leg is the geometric mean of the lengths of the hypotenuse and the segment on the hypotenuse which is adjacent to the leg. 2 = 20. Theorem: The product of the lengths of the legs of a right triangle is equal to the product of the lengths of the hypotenuse and altitude to this hypotenuse. c h b a bc = ah
10 PYTHGORS THORM 1. Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. OR In a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. b c a c 2 = a 2 + b 2 2. Theorem: If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. OR If in a triangle, square on one side is equal to the sum of the squares on the other two sides, then the angle opposite to the first side is right angle.
11 3. Theorem: In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is 3 times the length of the shorter leg. 3a 2a a 60 4. Theorem: The length of an altitude of an equilateral triangle with sides of length s is 3 s. 2 3 2 s s
12 5. Theorem: In a 45-45-90 triangle, the length of the hypotenuse is 2 times the length of a leg. a 2a
13 IRLS 1. efinition: secant is a line that intersects a circle in two points. O 2. Theorem: line that lies in the plane of a circle and contains an interior point of a circle is a secant. 3. efinition: tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point. The point of intersection is called the point of tangency. O P t
14 4. efinition: sphere is the set of all points in space that are a given distance from a given point. The given point is called the center of the sphere. Great ircles radius of a sphere is a segment determined by the center and a point on the sphere. diameter of a sphere is a segment that contains the center and has its endpoints on the sphere. The intersection of a sphere and a plane containing the center of the sphere is a great circle of the sphere.
15 Small circle: Intersection of a plane containing an interior point of the sphere, but not containing the center of the sphere. 5. Theorem: In a plane, a line is tangent to a circle if and only if it is perpendicular to a radius drawn to the point of tangency. OR tangent is perpendicular to the radius through the point of contact. OR line drawn at the end of a radius perpendicular to it is a tangent to the circle. O P t
16 6. Theorem: Segments drawn tangent to a circle from an exterior point are equal. = 7. efinition: common tangent is a line that is tangent to each of two coplanar circles. ommon xternal Tangent ommon external tangents do not intersect the segment joining the centers of circles. ommon internal tangents intersect the segment joining the centers of
17 circles. ommon Internal Tangent 8. efinition: Tangent circles are two coplanar circles that are tangent to the same line at the same point. tangent 9. efinition: circle is circumscribed about a polygon when the vertices of the polygon lie on the circle. The polygon is inscribed in the circle.
18 10. efinition: circle is inscribed in a polygon when the sides of the polygon are tangent to the circle. The polygon is circumscribed about the circle. 11. efinition: oncurrent lines are two or more lines that intersect in a single point. The point is called the point of concurrency. n m k 12. efinition: The circumcenter of a triangle is the point of concurrency of the perpendicular bisectors of the sides of the triangle. The circumcircle is
19 the circumscribed circle. IRUMNTR 13. Theorem: The angle bisector of a triangle are concurrent in a point equidistant from the sides of the triangle. 14. Theorem: circle can be inscribed in any triangle. 15. efinition: The incenter of a triangle is the point of concurrency of the angle bisectors of the sides of the triangle. The inscribed circle is called
20 incircle. G I F Inscribed ircle 16. efinition: The orthocenter of a triangle is the point of concurrency of the lines containing the altitudes the triangle. F H
21 17. efinition: The centroid of a triangle is the point of concurrency of the medians of the triangle. F 2 G 1 18. If two chords intersect in a circle, then the product of the lengths of the segments on one chord is equal to the product of the lengths of the segments on the other. O O O = O O
22 19. If two secant segments have a common endpoint in the exterior of a circle then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. O O O = O O 20. If a secant segment and a tangent segment have a common endpoint in the exterior of a circle then the product of the lengths of the secant segment and its external segment is equal to the square of the length of the tangent segment. P P P T = PT 2
23 R 1. Theorem: If two triangles are similar, then the ratio of their area is equal to the square of the ratio of the lengths of any two corresponding sides. F F area of area of F = 2 2 2. Theorem: If two polygons are similar, then the ratio of their area is equal to the square of the ratio of the lengths of any two corresponding sides. 3. efinition: sector of a circle is a region bounded by two radii and either the major arc or the minor arc that is intercepted. O
24 4. Theorem: In a circle of radius r, the ratio of the length s of an arc to the circumference of the circle is the same as the ratio of the arc measure m to 360. m s s m 360 5. efinition: segment of a circle is a region bounded by a chord and either the major arc or the minor arc that is intercepted. O 6. efinition: The ratio of the circumference of a circle to the diameter is denoted by.
25 c 1 c 2 d 1 d2 ny two circles are similar. Hence ircumference diameter 1 d d 7. Theorem: circle can be circumscribed about any regular polygon. 8. Theorem: circle can be inscribed about any regular polygon. 9. ircumference of a circle is the limit of the perimeters of its inscribed regular polygons as the number of sides increases. 1 2 2 rea of a circle is the limit of the area of its inscribed regular polygons as the number of sides increases.