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Similarity in Right Triangles

Chapter 7.4. Similarity in Right Triangles. Right Triangles. Leg. Leg. Altitude. Hypotenuse. The altitude is the Geometric Mean of the Segments of the Hypotenuse. Leg. Leg. Altitude. Short segment. Long segment. Short seg. =. Alt. Use the formula:. Alt. Long seg. Leg. Leg.

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Similarity in Right Triangles

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  1. Chapter 7.4 Similarity in Right Triangles

  2. Right Triangles Leg Leg Altitude Hypotenuse

  3. The altitude is the Geometric Mean of the Segments of the Hypotenuse Leg Leg Altitude Short segment Long segment

  4. Short seg = Alt • Use the formula: Alt Long seg Leg Leg Altitude Short segment Long segment

  5. Short seg • 2. Write the formula = Alt • 1. Label the triangle • Ex 1: Write the formula for the geometric mean Alt Long seg Leg Leg b a Altitude c e d Short segment Long segment

  6. Short seg c d = Alt • 3. Plug into the formula c e • Ex 1: Write the formula for the geometric mean Alt Long seg Leg Leg b a Altitude c e d Short segment Long segment

  7. Short seg z u = Alt z v • Ex 2: Write the formula for the geometric mean Alt Long seg Leg y Leg x Altitude z v u Short segment Long segment

  8. p Short seg n = = Alt • Ex 3: Write the formula for the geometric mean n q Alt Long seg Leg Leg k m Altitude n q p Short segment Long segment

  9. 4 Short seg 2 = = Alt 4 • Ex 4: Write the formula for the geometric mean 8 Alt Long seg Leg Leg k m Altitude 4 2 8 Short segment Long segment

  10. Short segment d • 1. Label the triangle Leg c • Ex 5: Write the formula for the geometric mean a Hypotenuse Altitude Long segment e Leg b

  11. Short segment d • 2. Write the formula Leg c • Ex 5: Write the formula for the geometric mean a Altitude Long segment e Leg b Short seg = Alt Alt Long seg

  12. Short segment d • 3. Plug into the formula Leg c • Ex 5: Write the formula for the geometric mean a Altitude Long segment e Leg b Short seg c d = Alt c e Alt Long seg

  13. The leg is the Geometric Mean between the whole Hypotenuse and the Segment of the Hypotenuse adjacent to the leg Leg Leg Altitude Hypotenuse

  14. Whole Hyp = Leg • Use the formula: Leg SegHypadj Leg Leg Altitude Short segment Long segment

  15. Remember – Adjacent sides share a common vertex Leg 2 Leg 1 Adjacent Hyp for leg 1

  16. Remember – Adjacent sides share a common vertex Leg 2 Leg 1 Adjacent Hyp for leg 2

  17. Whole Hyp • 2. Write the formula = Leg • 1. Label the triangle • Ex 6: Write the formula for the geometric mean using leg a. Leg SegHypadj Leg Leg b a c e d What is the whole hyp?

  18. What is the length of the whole hypotenuse? 4 9 12 y 5 2 3 x 2 + 4 = 6 3 + 12 = 15 5 + 9 = 14 x + y = x + y

  19. Whole Hyp a d+e = Leg • 3. Plug into the formula a • Ex 6: Write the formula for the geometric mean using leg a. d Leg SegHypadj Leg Leg b a Altitude c e d

  20. Whole Hyp x u+v = Leg • 3. Plug into the formula x u • Ex 7: Write the formula for the geometric mean using leg x. Leg SegHypadj Leg Leg y x Altitude z v u

  21. p+q Whole Hyp m = Leg • 3. Plug into the formula m p • Ex 8: Write the formula for the geometric mean using leg m. Leg SegHypadj Leg Leg k m Altitude n q p

  22. Whole Hyp 2+6 4 8 = Leg • 3. Plug into the formula 4 2 • Ex 9: Write the formula for the geometric mean using leg x. Leg SegHypadj Leg Leg 4 Altitude 2 6

  23. d • 1. Label the triangle Leg c • Ex 10: Write the formula for the geometric mean for side a. a Hypotenuse e Leg b

  24. d • 2. Write the formula Leg c • Ex 10: Write the formula for the geometric mean for side a. a e Leg b Whole Hyp = Side Side AdjHyp

  25. d • 3. Plug into the formula Leg c • Ex 10: Write the formula for the geometric mean for side a. a e Leg b Whole Hyp a d+e = Side a d Side AdjHyp

  26. Whole Hyp b d+e = Leg • 3. Plug into the formula • Ex 11: Write the formula for the geometric mean using leg b. b e Leg SegHypadj Leg Leg b a c e d

  27. 5+4 Whole Hyp 9 6 = Leg • 3. Plug into the formula 4 • Ex 11: Write the formula for the geometric mean using leg b. 6 Leg Seg Hyp adj Leg Leg 6 a c 4 5

  28. When to use which formula When you’re given the Altitude Short seg = Alt Alt Long seg Or When you need to find the Altitude

  29. When to use which formula When you’re given a Leg Whole Hyp Leg = Leg SegHypadj Or When you need to find a Leg

  30. Short segment d Leg • 3. Plug into the formula c • Ex 10: Write the formula for the geometric mean a Altitude Long segment e Leg b Short seg = Alt Alt Long seg

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