Chapter 7 Similarity.

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Presentation transcript:

Chapter 7 Similarity

7.1 Ratios and Proportions A comparison of two quantities A to B, A:B, or Proportion A Statement that two ratios are equal Ex.

Properties of Proportions If then 1 2 3 4

Cross-Product Property The product of the extremes is equal to the product of the means means extremes

Examples 7.1 Examples

7-2 Similar Polygons chapter 7.2 similarity.gsp Two polygons are similar if Corresponding angles are congruent Corresponding sides are proportional Similarity ratio Ratio of the lengths of the corresponding sides Practice with clickers

Golden Ratio http://www.youtube.com/watch?feature=player_detailpage&v=ReJOK8RMzPE http://www.youtube.com/watch?list=PL5E4F2F128AFE5A3D&feature=player_detailpage&v=085KSyQVb-U#t=4s

Golden Ratio 1:1.618 Also known as the golden rectangle

7.2 Examples Examples

7.3 Proving Triangles Similar AA Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Similarity Theorems SAS Similarity SSS Similarity If an angle of one triangle is congruent to an angle of a second triangle, and the sides including the two angles are proportional, then the triangles are similar. SSS Similarity If the corresponding sides of two triangles are proportional, then the triangles are similar.

Examples Examples

7.4 Similarity in Right Triangles Theorem 7-3 The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.

Geometric Mean The geometric mean of a and b is

Example Find the geometric mean of 4 and 18

Corollaries to Theorem 7-3 Corollary 1 Piece of hypotenuse = Altitude Altitude Other piece of hypotenuse Corollary 2 Piece of hypotenuse = Leg Leg Whole Hypotenuse

7.5 Proportions in Triangles Theorem 7-4 Side Splitter Theorem If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. =

Corollary to Theorem 7-4 If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional. c a d d b

Theorem 7-5 example Triangle-Angle-Bisector Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. p m o n