1. Geometry 9.2 Special Right Triangles
9.2 Day 1 Warm Up
Find the value of x. Give answers as simplified radicals. x x 3 x
February 27, 2019
Geometry 9.2 Special Right Triangles

2. 9.2 Special Right Triangles
Geometry
9.2 Special Right Triangles

3. Geometry 9.2 Special Right Triangles
9.2 Essential Question
What is the relationship among the side lengths of 45°- 45°- 90° triangles? °- 60°- 90° triangles?
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Geometry 9.2 Special Right Triangles

4. Geometry 9.2 Special Right Triangles
Goals
Know the side lengths of special right triangles.
Use special right triangles to solve problems.
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Geometry 9.2 Special Right Triangles

5. Geometry 9.4 Special Right Triangles
First, radical review:
A radical in simplest form has no perfect squares in the radicand.
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Geometry 9.4 Special Right Triangles

6. Rationalizing Denominators
This means no square roots allowed in the denominator of fractions.
General rule is to multiply numerator and denominator by the radical and simplify.
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Geometry 9.2 Special Right Triangles

7. Geometry 9.2 Special Right Triangles
Examples
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Geometry 9.2 Special Right Triangles

8. Geometry 9.2 Special Right Triangles
February 27, 2019
Geometry 9.2 Special Right Triangles

9. Constructing Special Triangles
Given a Square.
Draw one diagonal.
The diagonal bisects the angles.
1 & 2 measure ______. 2 1
45
February 27, 2019
Geometry 9.2 Special Right Triangles

10. Constructing Special Triangles
Clean it up – keep only the triangle on the right.
45
45
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Geometry 9.2 Special Right Triangles

11. Constructing Special Triangles
In the original square, each side was the same. Label them a.
45 a
45 a
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Geometry 9.2 Special Right Triangles

12. Constructing Special Triangles
Solve for the hypotenuse, c.
45
𝑎 2 c a
45 a
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Geometry 9.2 Special Right Triangles

13. Geometry 9.2 Special Right Triangles
Triangle Theorem (Thm 9.4)
In a 45°- 45°- 90° triangle, the hypotenuse is 2 times as long as each leg.
45 a
𝑎 2
February 27, 2019
Geometry 9.2 Special Right Triangles

14. Geometry 9.2 Special Right Triangles
Example 1
Solve for x & y.
45
45 a
a2 y 5 x
52 a
45 5
February 27, 2019
Geometry 9.2 Special Right Triangles

15. Geometry 9.2 Special Right Triangles
Example 2
Solve for x & y.
45
45 a
a2 x
82 8 a
45 8 y
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Geometry 9.2 Special Right Triangles

16. Geometry 9.2 Special Right Triangles
Example 3: A tougher one.
Solve for x & y.
45
45 a
a2 10 x a
45 y
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Geometry 9.2 Special Right Triangles

17. Geometry 9.2 Special Right Triangles
Example 3 Solution
45 10 x
52
45 y
52
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Geometry 9.2 Special Right Triangles

18. Geometry 9.2 Special Right Triangles
Learn the pattern!
45 a
a2
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Geometry 9.2 Special Right Triangles

19. Geometry 9.2 Special Right Triangles
Try it. Find x and y.
45 8 y
42
45 x
42
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Geometry 9.2 Special Right Triangles

20. Geometry 9.2 Special Right Triangles
Shortcut pattern:
45 2a
𝑎 2
45
𝑎 2
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Geometry 9.2 Special Right Triangles

21. Geometry 9.2 Special Right Triangles
Example 4 Solve for x & y 40 y
202 x
202
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Geometry 9.2 Special Right Triangles

22. Geometry 9.2 Special Right Triangles
Example 5
A designer wants to put a rope light on the diagonal of a square dance floor. If the floor measures 30 ft. on a side, how long does the rope light need to be?
February 27, 2019
Geometry 9.2 Special Right Triangles

23. Geometry 9.2 Special Right Triangles
Example 5 Solution
302  ft.  42 ft 5 in.
45
302
30 ft.
45
30 ft.
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Geometry 9.2 Special Right Triangles

24. Geometry 9.2 Special Right Triangles
9.2 Day 2 Warm Up
Solve for the variable X 12
4 3 8 X 4
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Geometry 9.2 Special Right Triangles

25. Constructing Special Triangles
Construct an equilateral triangle.
Each angle measures ______.
60
60
60
60
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Geometry 9.2 Special Right Triangles

26. Constructing Special Triangles
Draw an altitude.
It is perpendicular to the base.
It also bisects the vertex angle.
The altitude divides the triangle into two congruent triangles with angles measures of
60
30
60
60
February 27, 2019
Geometry 9.2 Special Right Triangles

27. Constructing Special Triangles
Clean up the drawing – only keep the triangle on the left.
30
60
60
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Geometry 9.2 Special Right Triangles

28. Constructing Special Triangles
This is called a 30 – 60 – 90 triangle.
30
60
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Geometry 9.2 Special Right Triangles

29. Constructing Special Triangles
Give the original side an arbitrary length. Call it 2a.
30 2a
60
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Geometry 9.2 Special Right Triangles

30. Constructing Special Triangles
What is the length of the base?
30 2a
60 a ? 2a
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Geometry 9.2 Special Right Triangles

31. Constructing Special Triangles
Now find the height of the triangle, h.
a2 + h2 = (2a)2
30 2a h
60 a
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Geometry 9.2 Special Right Triangles

32. Constructing Special Triangles
30 2a h
60 a
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Geometry 9.2 Special Right Triangles

33. Geometry 9.2 Special Right Triangles
60
30 2a a
Short leg
Long
leg
Hypotenuse
Triangle Theorem (Theorem 9.5)
In a 30°- 60°- 90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is 3 times as long as the shorter leg.
February 27, 2019
Geometry 9.2 Special Right Triangles

34. Geometry 9.2 Special Right Triangles
60
30
60
30
2a 3
a 3
60
30 2a a
3𝑎 2
𝑎 3 2 3a
a 3
February 27, 2019
Geometry 9.2 Special Right Triangles

35. Geometry 9.2 Special Right Triangles
Example 6
Find x & y.
30
60
30 2a a 12 y
63
60 x 6
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Geometry 9.2 Special Right Triangles

36. Geometry 9.2 Special Right Triangles
Example 7
Find x & y.
30
60
30 2a a 16
83 x y
60 8
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Geometry 9.2 Special Right Triangles

37. Geometry 9.2 Special Right Triangles
Example 8
Find x & y.
30
60
30 2a a y
20 3 40
60 x 20
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Geometry 9.2 Special Right Triangles

38. Geometry 9.2 Special Right Triangles
Example 9
Find x & y.
30 y 12
60 x
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Geometry 9.2 Special Right Triangles

39. Geometry 9.2 Special Right Triangles
Example 9 Solution
30 y 12
60 x
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Geometry 9.2 Special Right Triangles

40. Geometry 9.2 Special Right Triangles
Learn the pattern!
30 2a
60 a
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Geometry 9.2 Special Right Triangles

41. Geometry 9.2 Special Right Triangles
Try it. Find x and y.
30 y x
153 30
60 15
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Geometry 9.2 Special Right Triangles

42. Another one: Find h and y.
30 h
73 14
60 y 7
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Geometry 9.2 Special Right Triangles

43. Geometry 9.2 Special Right Triangles
Again: Find x and y.
30 x 4
2 𝟑
60 y 2
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Geometry 9.2 Special Right Triangles

44. Geometry 9.2 Special Right Triangles
Summary
60
30 2a a
45 a
a 2
In Trigonometry, you will understand why these triangle are so important.
February 27, 2019
Geometry 9.2 Special Right Triangles

45. Geometry 9.2 Special Right Triangles
Quick Quiz – 5 problems
30 x y 12
1. Find x & y.
123 24
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Geometry 9.2 Special Right Triangles

46. Geometry 9.2 Special Right Triangles
2. Find x & y.
45
7 2 y x
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Geometry 9.2 Special Right Triangles

47. Geometry 9.2 Special Right Triangles
3. Find x & y.
60 50 x 25 y
253
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Geometry 9.2 Special Right Triangles

48. Geometry 9.2 Special Right Triangles
4. Find x & y. 62 x
45 y
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Geometry 9.2 Special Right Triangles

49. Geometry 9.2 Special Right Triangles
5. Find x & y. 75
60 x y
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Geometry 9.2 Special Right Triangles

50. Geometry 9.2 Special Right Triangles
6. Find x & y.
60 y
9 3 x
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Geometry 9.2 Special Right Triangles

51. Geometry 9.2 Special Right Triangles
One More Time…
60
30 2a a
45 a
a2
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Geometry 9.2 Special Right Triangles

52. Geometry 9.2 Special Right Triangles
Homework
February 27, 2019
Geometry 9.2 Special Right Triangles