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Unit 10: Similarity (7-2) Similar Polygons Similar figures – Similar Polygons – two polygons are similar polygons if corresponding angles are congruent and if the lengths of corresponding sides are proportional ABCD ~ GHIJ Diagram: Scale factor – Understanding Similarity Example 1: ∆MNP ~ ∆SRT a) Congruent angles? b) Extended proportion for the ratios of corresponding sides? Unit 10: Similarity Example 2: DEFG ~ HJKL a) Congruent angles? b) Extended proportion for ratios? Determining Similarity Example 3: Are the polygons similar? JK = TU KL = UV LM VW JM TW Example 4: = = Are the polygons similar? AB DE BC EF AC DF = = = Unit 10: Similarity Using Similar Polygons Example 5: ABCD ~ EFGD. What is the value of x? What is the value of y? Scale Drawing Scale drawing – Scale – the ratio that compares each length in the scale drawing to the actual length/ Example 6: The length of a bridge in a drawing is 6.4 cm. The scale drawing is 1 cm = 200 m. Extra Example: Problem 4, Page 443 Unit 10: Similarity (7-3) Proving Triangles Similar Angle – Angle Similarity (AA ~ ) Postulate If two angles from one triangle are congruent to two angles of another, then triangles are similar ∆SRT ~ ∆MLP Diagram: Example 1: Are the triangles similar? Example 2: Are the triangles similar? Example 3: Are the triangles similar? Unit 10: Similarity Side-Angle-Side Similarity (SAS ~ ) Theorem Contains one pair of congruent angles; Sides that include the two angles are proportional Diagram: If… AB QR = AC QS and ∠A ≅ ∠Q Then… ∆ABC ~ ∆QRS Example 4: Are the triangles in this figure similar? Side-Side-Side Similarity (SSS ~ ) Theorem Corresponding sides are proportional Diagram: If… AB QR = AC QS = BC RS Then… ∆ABC ~ ∆QRS Unit 10: Similarity Example 5: Are the triangles similar? Shortest sides: Longest sides: ST XV US WX Remaining sides: = = TU VW = Indirect Measurement - Uses the fact that light reflects off a mirror at the same angle at which it hits the mirror Example 6: Before rock climbing, Darius wants to know how high he will climb. He places a mirror on the ground and walks backward until he can see the top of the cliff in the mirror. What is the height of the cliff? Unit 10: Similarity (7-4) Similarity in Right Triangles Theorem 7-3 The altitude to the hypotenuse of a right triangle divides the triangle into 2 similar triangles that are also similar to the original Example 1: Write the similarity statement relating the three triangles. Geometric Mean For any two positive numbers a and b, the geometric mean of a and b is the positive number x a x such that = x b Example 2: What is the geometric mean of 6 and 15? Example 3: What is the geometric mean of 4 and 18? Unit 10: Similarity Corollary 1 to Theorem 7-3 and Corollary 2 to Theorem 7-3 Unit 10: Similarity (Concept Byte) The Golden Ratio – Page 468 Unit 10: Similarity (7-5) Proportions in Triangles Side-Splitter Theorem If ⃡ RS Then… Example 1: What is the value of x? Example 2: What is the value of a? ∥ ⃡ XY XR RQ = YS SQ Unit 10: Similarity Corollary to Side-Splitter Theorem If a ∥b∥c Then… Example 3: AB BC = WX XY Problem 2, Page 473: Three campsites are shown. What is the length of Site A? Diagram: Let x be the length of Site A. Example 4: What is the length of Site C? Let y be the length of Site C. Unit 10: Similarity Triangle-Angle-Bisector Theorem If AD bisects ∠ CAB Then… Example 5: What is the value of x? Example 6: What is the value of y? CD DB = CA BA