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Chapter Seven Similar Polygons Ruby Weiner & Leigh Zilber 7.1 Ratios and Proportions • Ratio: the quotient of 2 values D A 60 60 E 90 90 30 C 30 B 1) Find the ratio of AE to BE 10: 5x 2:x 1) Find the ratio of largest > of triACE to smallest > of triBDE 90:30 3:1 Ratio Practice Problems • A telephone pole 7 meters tall snaps into 2 parts. The ratio of the 2 parts is 3:2. Find the length of each part. • A teams best hitter has a life time batting average of .320. He has been at bat 325 times. – how many hits has he made? workout 1) 3x + 2x = 7 5x = 7 --> 7/5 3(7/5) = 21/5 meters 2(7/5) = 14/5 meters 2) x/325 = 32/100 100x = 325 x 32 100x = 10400 x = 104 4 7.2 Properties of Proportions • Proportion: equation stating that 2 ratios are equal • Properties (given a/b = c/d) : – b/a = d/c – ad = bc – a/c = b/d – a+b/b = c+d/d 5 examples: (given a/b = 3/5) 1. 5a = 3b 2. 5/b = 3/a 3. a+b/b = 3+5/5 --> 8/5 4. 5/3 = b/a Proportion Practice Problems • Choose yes or no – given: 10/20 = a/b – – – – – 6 is 10 x b = 20 x a ? is 10/20 = b/a ? is 30/20 = a+b/b ? is 20/10 = b/a ? 10/a = b/20? ANSWERS YES b.c ad = bc NO b.c a/b no= d/c YES b.c a+b/b = c+d/d YES b.c b/a = d/c NO b.c a/c no= d/b 7 7.3 Similar Polygons • recall congruent triangles – corresponding angles --> congruent – corresponding sides --> congruent Similar triangles E B A 8 C D -Corresponding angles are congruent -Corresponding sides are in proportion -AB/DE = BC/EF = AC/DF F Examples: find length of EF if triABC is similar to triDEF E B 2 A 4 4 3 C D 2/4 = 4/x 2x = 16 x=8 9 x 6 F 7-4 A Postulate for Similar Triangles • Postulate 15: AA Similarity Postulate - If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Example: Are these triangle similar? How? B A Conclude: yes, AA Similarity (AA~) C Practice Problems o Determine if the triangles are similar and how. 1) 50 40 2) Given: Both Triangles are Isosceles 5 5 5 5 Answers • 1. 40 + 90 + x = 180 x = 50 50 + 90 + x = 180 x = 40 they are similar by AA similarity 7-5 Theorems for Similar Triangles • Theorem 7-1 (SAS Similarity Theorem) - If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar. A Example: Are these Triangles congruent? Why? Answer: Yes, SAS~B Given: Angle A is congruent to Angle D AB/DE = AC/DF x C E 1 y F 7-5 Continued • Theorem 7-2 (SSS Similarity Theorem) - If the sides of two triangles are in proportion, then the triangles are similar. Example: A D Answer: Yes, SSS~ X Given: AB/DE = BC/EF = AC/DF B C E 1 Y F Practice Problems E • 1. 10 6 C B 15 D 9 A • 2. L 8 O 7.5 5 N 65 K P 65 12 M Answers • 1. Triangle BAC ~ Triangle EDC; SAS~ • 2. Triangle LKM ~ Triangle NPO; SAS~ 7-6 Proportional Lengths • Theorem 7-3 (Triangle Proportionality Theorem) - If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. Example: Find the numerical value of R A) TN/ NR B) TR/NR 6 C) RN/RT M N Answer: a) tn/nr = sm/mr = 3/6 = ½ b) Tr/nr = sr/mr = 9/6 = 3/2 c) Rn/rt = rm/rs = 6/9 = 2/3 3 S T 7-6 Continued • Corollary - If three parallel lines intersect two transversals, then they divide the transversals proportionally. • Theorem 7-4 (Triangle Bisector Theorem) - If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides. 7-6 Continued Example: F G 2 D Given: Triangle DEF; Ray DG bisects Angle FDE Prove: GF/ GE = DF/DE 1 3 4 K E Answer: (Plan for Proof) Draw a line through E parallel to Ray DG And intersecting Ray FD at K. Apply Triangle Proportionality Theorem To Triangle FKE. Triangle DEK is isosceles With DK = DE. Substitute this into your Proportion to complete the proof. Practice Problems 1. State a proportion for the diagram: n a g b Answer • 1. a/n = b/g Practice Proof • Given: Angle H and Angle F are right triangles K Prove: HK * GO = FG * KO Statements Reasons H 1 O 2 F G Answer to Proof Statements 1. Angle 1 is congruent to Angle 2. 2. Angle H and Angle F are right Triangles. 3. Angle H = 90 and Angle F = 90 4. Angle H is congruent to Angle F. 5. Triangle HKO ~ Triangle FGO 6. HK/FG = KO/GO 7. HK*GO = FG*KO Reasons 1. Vertical Triangles are congruent. 2. Given 3. Def. of right triangle. 4. Def. of congruent triangle 5. AA~ 6. Corr. Sides of ~ Triangles are in proportion. 7. A property of proportions.