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Chapter Seven Similar Polygons

Chapter Seven Similar Polygons. Ruby Weiner & Leigh Zilber. 7.1 Ratios and Proportions. Ratio: the quotient of 2 values. D. A. 60. 60. 10. E. 90. 90. Find the ratio of AE to BE 10: 5x  2:x Find the ratio of largest > of triACE to smallest > of triBDE 90:30  3:1. 30. C.

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Chapter Seven Similar Polygons

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  1. Chapter SevenSimilar Polygons Ruby Weiner & Leigh Zilber

  2. 7.1 Ratios and Proportions • Ratio: the quotient of 2 values D A 60 60 10 E 90 90 • Find the ratio of AE to BE • 10: 5x 2:x • Find the ratio of largest > of triACE to smallest > of triBDE • 90:30 3:1 30 C 5x 30 B

  3. 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?

  4. 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

  5. 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 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

  6. Proportion Practice Problems • Choose yes or no • given: 10/20 = a/b • 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?

  7. 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

  8. 7.3 Similar Polygons • recall congruent triangles • corresponding angles --> congruent • corresponding sides --> congruent Similar triangles E -Corresponding angles are congruent -Corresponding sides are in proportion -AB/DE = BC/EF = AC/DF B A C D F

  9. Examples: find length of EF if triABC is similar to triDEF E B 4 2 4 x A C 3 D F 6 2/4 = 4/x 2x = 16 x = 8

  10. 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? Conclude: yes, AA Similarity (AA~) B A C

  11. Practice Problems • Determine if the triangles are similar and how. 1) 2) Given: Both Triangles are Isosceles 50 40 5 5 5 5

  12. Answers • 1. 40 + 90 + x = 180 x = 50 50 + 90 + x = 180 x = 40 they are similar by AA similarity

  13. 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. Example: Are these Triangles congruent? Why? Answer: Yes, SAS~ A 1 x y B C F E Given: Angle A is congruent to Angle D AB/DE = AC/DF

  14. 7-5 Continued • Theorem 7-2 (SSS Similarity Theorem) • If the sides of two triangles are in proportion, then the triangles are similar. Example: Answer: Yes, SSS~ A D 1 Y X Given: AB/DE = BC/EF = AC/DF C E F B

  15. Practice Problems E • 1. • 2. 10 6 C B 15 D 9 A P L 65 8 5 O 7.5 N 65 K 12 M

  16. Answers • 1. Triangle BAC ~ Triangle EDC; SAS~ • 2. Triangle LKM ~ Triangle NPO; SAS~

  17. 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 A) TN/ NR B) TR/NR C) RN/RT R 6 M N Answer: tn/nr = sm/mr = 3/6 = ½ Tr/nr = sr/mr = 9/6 = 3/2 Rn/rt = rm/rs = 6/9 = 2/3 3 S T

  18. 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.

  19. 7-6 Continued Example: F G 2 1 D E 3 Given: Triangle DEF; Ray DG bisects Angle FDE Prove: GF/ GE = DF/DE 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. 4 K

  20. Practice Problems State a proportion for the diagram: n a g b

  21. Answer • 1. a/n = b/g

  22. Practice Proof • Given: Angle H and Angle F are right triangles Prove: HK * GO = FG * KO K Statements Reasons H 1 O 2 G F

  23. Answer to Proof Statements Reasons Angle 1 is congruent to Angle 2. Angle H and Angle F are right Triangles. Angle H = 90 and Angle F = 90 Angle H is congruent to Angle F. Triangle HKO ~ Triangle FGO HK/FG = KO/GO HK*GO = FG*KO Vertical Triangles are congruent. Given Def. of right triangle. Def. of congruent triangle AA~ Corr. Sides of ~ Triangles are in proportion. A property of proportions.

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