1. 9.3 Warmup Find the value of x and y. 1. 2. 3.4 x 10
6. Are these the sides of a triangle?
If yes, is the
acute, obtuse or
right?
a. 4, 4, 10
b. 9, 15,12
c. 2, 3, 4 𝑥 36 𝑦
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Geometry 9.1 Similar Right Triangles

2. 9.3 Similar Right Triangles
Geometry
9.3 Similar Right Triangles
9.3
Work
Sheet

3. Geometry 9.3 Similar Right Triangles
9.3 Essential Question
How are altitudes and geometric means of right triangles related?
January 18, 2019
Geometry 9.3 Similar Right Triangles

4. Geometry 9.3 Similar Right Triangles
Goals
Know proportions in similar right triangles.
Solve problems involving similar right triangles formed by altitudes on the hypotenuse.
January 18, 2019
Geometry 9.3 Similar Right Triangles

5. Geometry 9.3 Similar Right Triangles
Means (Averages)
Arithmetic mean of x & y:
Geometric mean of x & y:
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Geometry 9.3 Similar Right Triangles

6. Geometry 9.3 Similar Right Triangles
Geometric Mean
The geometric mean of two positive numbers a and b is the positive number x that satisfies 𝑎 𝒙 = 𝒙 𝑏 .
So, 𝒙 2 =𝑎𝑏 and 𝒙= 𝑎𝑏 .
January 18, 2019
Geometry 9.3 Similar Right Triangles

7. Geometry 9.3 Similar Right Triangles
Example 1
a. Find the geometric mean of 24 and 48. b. Find the geometric mean of 18 and 54.
January 18, 2019
Geometry 9.3 Similar Right Triangles

8. Geometry 9.3 Similar Right Triangles
Your Turn
a. Find the geometric mean of 12 and 27. b. Find the geometric mean of 16 and 18.
January 18, 2019
Geometry 9.3 Similar Right Triangles

9. Remember AA~ for Triangles
Theorem: If two angles of one triangle are congruent to two angles of a another triangle, then the triangles are similar.
(AA~ Postulate)
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Geometry 9.3 Similar Right Triangles

10. Right Triangle Similarity Theorem
Start with right ABC with altitude 𝐶𝐷 ( CD  AB at D). C B A D
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Geometry 9.3 Similar Right Triangles

11. Right Triangle Similarity TheoremC B A D
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Geometry 9.3 Similar Right Triangles

12. Right Triangle Similarity TheoremC D B
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Geometry 9.3 Similar Right Triangles

13. Right Triangle Similarity TheoremC
If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. C A C D B D B A
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Geometry 9.3 Similar Right Triangles

14. Right Triangle Similarity TheoremC
ABC ~ ACD ~ CBD
(AA~ Postulate) C A C D B D B A
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Geometry 9.3 Similar Right Triangles

15. Right Triangle Similarity Theorem
ABC ~ ACD ~ CBD C
For clarity, we name the segments. b a h B A y D x c
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Geometry 9.3 Similar Right Triangles

16. Geometry 9.3 Similar Right Triangles
ABC ~ ACD ~ CBD C
𝑎 𝑏 = 𝒉 𝑦 = 𝑥 𝒉 ⇒
ℎ 2 =𝑥𝑦
ℎ= 𝑥𝑦 a h C A C a b c D x b B h y D B A
January 18, 2019
Geometry 9.3 Similar Right Triangles

17. Geometry 9.3 Similar Right Triangles
ABC ~ ACD ~ CBD C
𝒂 𝑐 = ℎ 𝑏 = 𝑥 𝒂 ⟹
𝑎 2 =𝑥𝑐
𝑎= 𝑥𝑐 a h C A C a b c D x b B h y D B A
January 18, 2019
Geometry 9.3 Similar Right Triangles

18. Geometry 9.3 Similar Right Triangles
ABC ~ ACD ~ CBD C
𝒃 𝑐 = 𝑦 𝒃 = ℎ 𝑎 ⇒
𝑏 2 =𝑦𝑐
𝑏= 𝑦𝑐 a h C A C a b c D x b B h y D B A
January 18, 2019
Geometry 9.3 Similar Right Triangles

19. Geometry 9.3 Similar Right Triangles
Similar Triangles
ABC ~ ACD ~ CBD
These expressions are called geometric means.
ℎ= 𝑥𝑦 𝑎= 𝑥𝑐 𝑏= 𝑦𝑐 A C D a b x y h c B
January 18, 2019
Geometry 9.3 Similar Right Triangles

20. Theorem 9.7 Geometric Mean (Altitude) Thm
The altitude drawn to the hypotenuse of a right triangle is the geometric mean of the segments on the hypotenuse.
𝒉= 𝒙𝒚 h y x
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Geometry 9.3 Similar Right Triangles

21. Geometry 9.3 Similar Right Triangles
Example 2 Find h.
ℎ= 𝑥𝑦 h 6 9 4
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Geometry 9.3 Similar Right Triangles

22. Geometry 9.3 Similar Right Triangles
Your Turn Find x.
ℎ= 𝑥𝑦 4 2 8 x
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Geometry 9.3 Similar Right Triangles

23. Theorem 9.8 Geometric Mean (Leg) Thm
The length of each leg of a right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg.
𝒂= 𝒙𝒄
𝒃= 𝒚𝒄 b a y x c
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Geometry 9.3 Similar Right Triangles

24. Geometry 9.3 Similar Right Triangles
Example 3 Find a & b.
𝑎= 𝑥𝑐
𝑏= 𝑦𝑐 b a 10 5 15
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Geometry 9.3 Similar Right Triangles

25. Geometry 9.3 Similar Right Triangles
Your Turn Find a & b.
𝑎= 𝑥𝑐 𝑏= 𝑦𝑐 b a 8 2
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Geometry 9.3 Similar Right Triangles

26. Geometry 9.3 Similar Right Triangles
All you need to know…
ABC ~ ACD ~ CBD
ℎ= 𝑥𝑦 𝑎= 𝑥𝑐 𝑏= 𝑦𝑐 D a b x y h c
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Geometry 9.3 Similar Right Triangles

27. Quickly Complete the Equation6 x t
𝐲= 𝟒 × 𝟏𝟎 4 y
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Geometry 9.3 Similar Right Triangles

28. Quickly Complete the Equation
𝑥= 7×11
𝑡= 7×4 7 x t 4 y
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Geometry 9.3 Similar Right Triangles

29. Example 4: Solve for a, b, & c.6 x 4 c
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Geometry 9.3 Similar Right Triangles

30. Geometry 9.3 Similar Right Triangles
Solution a b 6 x 4 c
Begin here
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Geometry 9.3 Similar Right Triangles

31. Geometry 9.3 Similar Right Triangles
Solution a b 6 9 4 13 c
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Geometry 9.3 Similar Right Triangles

32. Geometry 9.3 Similar Right Triangles
Solution
10.82 b 6 9 4 13
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Geometry 9.3 Similar Right Triangles

33. Geometry 9.3 Similar Right Triangles
Solution
10.82
7.21 6 9 4 13
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Geometry 9.3 Similar Right Triangles

34. Geometry 9.3 Similar Right Triangles
This is hard!
Open your eyes!
No, it isn’t.
Ask: what segment do you want to find?
Which others do you need to know?
Which formula form?
Solve the formula for the missing segment.
January 18, 2019
Geometry 9.3 Similar Right Triangles

35. Geometry 9.3 Similar Right Triangles
All you need to know…
ABC ~ ACD ~ CBD
ℎ= 𝑥𝑦 𝑎= 𝑥𝑐 𝑏= 𝑦𝑐 D a b x y h c
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Geometry 9.3 Similar Right Triangles

36. Geometry 9.3 Similar Right Triangles
You do it. Solve for a, b, h. b a 6 h 15 21
January 18, 2019
Geometry 9.3 Similar Right Triangles

37. Geometry 9.3 Similar Right Triangles
Try this. Solve for d, e, & f. d e 6 f
4.5
6= 4.5 𝑓
36=4.5 𝑓
𝑓=8
January 18, 2019
Geometry 9.3 Similar Right Triangles

38. Geometry 9.3 Similar Right Triangles
Try this. Solve for d, e, & f. d e 6 8
4.5
12.5
d= 12.5(8 )
𝑑= 100
𝑑=10
e= 12.5(4.5 ) 𝑑=
𝑑=7.5
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Geometry 9.3 Similar Right Triangles

39. Geometry 9.3 Similar Right Triangles
Is this possible? 9 h 16 5
No: 9 ≠ 8.94
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Geometry 9.3 Similar Right Triangles

40. Geometry 9.3 Similar Right Triangles
Homework
January 18, 2019
Geometry 9.3 Similar Right Triangles