1. REVIEW OF ROOTS
This is a collection of warm-ups and practice from class.
Click to advance the slide and follow along.
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2. Make sure you have an answer!
WARM-UP
What is the prime factorization of the following numbers (ex. 14=2 • 7):
9, 16, 25, 10, 17, 24, 27
What are the square roots of the following numbers:
Make sure you have an answer!

3. Answers 9 = 3 9 = 3•3 16 = 4 16 = 2•2•2•2 25 = 5 25 = 5•5
10 = 10 or 3.2
17 = 17 or 4.1
24 = 2 6 or 4.9
27 = 3 3 or 5.2
9 = 3•3
16 = 2•2•2•2
25 = 5•5
10 = 2•5
17 = 17
24 = 2•2•2•3
27 = 3•3•3

4. Chapter 10 – Right Triangles
Why should you care?
LOTS OF STANDARDS: Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles.
Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.
Students know and are able to use angle and side relationships in problems with special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90° triangles.

5. Chapter 10 – Right Triangles
Why should you care?
TRIGONOMETRY: You’ll need this stuff for next year!!
Mr. Taylor’s Opinion. Pythagorean theorem and the trigonometric functions are EXTREMELY useful for DOING practical problems involving graphing(drawing) and measurement.

6. ROOTS – From Chapter 3 WHY NOT USE A CALCULATOR?
What is the square root of 5 on a calculator?
approximately 2.236
What’s the square root of 5 squared? 5
What’s the squared?
(close to 5 but not exactly)
THEREFORE, if we want exactly the square root of 5, use

7. Finding exact roots
To simplify a number which includes a radical, find the prime factorization of the radicand and move all the perfect squares out front.
Examples. 4, 6, 8, 12, 15, 18

8. Pythagorean Theorem
The Pythagorean Theorem states that for any triangles with legs a and b and hypotenuse c that a2 + b2 = c2
If a right triangle has legs 3 and 4, what is the length of the hypotenuse?
If a right triangle has a leg 2 and a hypotenuse 10, what is the length of the other leg?
If a triangle has sides 5, 6, and 8 is it a right triangle?

9. Pythagorean Theorem 25 = c2 32 + 42 = c2 9 + 16 = c2 25 = c2 5 = c
The Pythagorean Theorem states that for any triangles with legs a and b and hypotenuse c that a2 + b2 = c2
If a right triangle has legs 3 and 4, what is the length of the hypotenuse?
= c2
= c2
25 = c2
25 = c2
5 = c

10. Pythagorean Theorem b2 = 6 22 + b2 = (10)2 4 + b2 = 10 b2 = 10 – 4
The Pythagorean Theorem states that for any triangles with legs a and b and hypotenuse c that a2 + b2 = c2
2. If a right triangle has a leg 2 and a hypotenuse 10, what is the length of the other leg?
22 + b2 = (10)2
4 + b2 = 10
b2 = 10 – 4
b2 = 6
b2 = 6
b = 6

11. Pythagorean Theorem 52 + 62 = 82 25 + 36 = 64 51  64
The Pythagorean Theorem states that for any triangles with legs a and b and hypotenuse c that a2 + b2 = c2
2. If a triangle has sides 5, 6, and 8 is it a right triangle?
= 82
= 64
51  64
NO, Not a right triangle

12. WARM-UP What are the lengths of the missing sides? 60 45 12 72 6 7 9030 45
62 + b2 = 122
b2 = 144 – 36
b =  108
b = 6  3
63
= c2
c2 = 49+49
c = 2(7)(7)
c = 7  2 7

13. Gold Boxes from p. 424
In a Triangle, the measure of the hypotenuse is 2 times the leg. 45
x2
x2 + x2 = c2
c2 = x2 + x2
c2 = 2x2
c = 2x2
c = x 2 x 90 45 x

14. 45-45-90 is an isosceles triangle
What about this? 45
72 2 = 7 * 2 = 14
72 90 45
72

15. 45-45-90 is an isosceles triangle
76 2 45
= 72*2*3 = 7 * 2 * 3 = 143
76 90 45
76

16. 45-45-90 is an isosceles triangle
32 3 90 45 3

17. 45-45-90 is an isosceles triangle7 90 45
This answer
Would never
Be on a M.C.
Test
Better 

18. 45-45-90 is an isosceles triangle8 90 45
Better 

19. Preview 30/60/90

20. 30-60-90 is half an equilateral triangle
(Assume short side is opposite small angle)
42 + X2 = 82
X2 = 64 – 16
X =  48
X = 4  3 60 8 4 90 30
43

21. 30-60-90 is half an equilateral triangle
72 + X2 = 142
X2 = 196 – 49
X =  147
X = 7  3 60 14 7 90 30
73

22. 30-60-90 is half an equilateral triangle
62 + X2 = 122
X2 = 144 – 36
X =  108
X = 6  3 60 12 6 90 30
63

23. 30-60-90 is half an equilateral triangle
m2 + X2 = (2m)2
X2 = 4m2 – m2
X2 = 3m2
X = m  3 60 2m m 90 30
m3

24. WARM-UP What are the lengths of the missing sides? 60 45 10 92 5 9 9030 45
52 + b2 = 102
b2 = 100 – 25
b =  75
b = 5  3
53
= c2
c2 = 81+81
c = 2(9)(9)
c = 9  2 9

25. 45-45-90 is an isosceles triangle
92 9 90 45 9

26. 45-45-90 is an isosceles triangle
52 5 90 45 5

27. 45-45-90 is an isosceles triangle
62 2 = 6 * 2 = 12
62 90 45
62

28. 45-45-90 is an isosceles triangle
76 2 = 7 * 2 * 3 = 143
76 90 45
76

29. 45-45-90 is an isosceles triangle
32 3 90 45 3

30. 45-45-90 is an isosceles triangle7 90 45
This answer
Would never
Be on a M.C.
Test
Better 

31. 45-45-90 is an isosceles triangle10 90 45
Better 

32. 30-60-90 is half an equilateral triangle
(Assume short side is opposite small angle)
42 + X2 = 82
X2 = 64 – 16
X =  48
X = 4  3 60 8 4 90 30
43

33. 30-60-90 is half an equilateral triangle
72 + X2 = 142
X2 = 196 – 49
X =  147
X = 7  3 60 14 7 90 30
73

34. 30-60-90 is half an equilateral triangle
m2 + X2 = (2m)2
X2 = 4m2 – m2
X2 = 3m2
X = m  3 60 2m m 90 30
m3

35. Rules:
In a Triangle, the measure of the hypotenuse is the leg times 2
In a Triangle:
hypotenuse = 2 x shorter leg
longer leg = 3 x shorter leg
To go in reverse direction, reverse the operation.
For instance, to go from hypotenuse to leg in a , divide by 2
Answers must have no perfect squares under the radicals and no radicals in the denominator.

36. 30-60-90 is half an equilateral triangle
426
213 90 30
2133

37. 30-60-90 is half an equilateral triangle2 1 90 30 3

38. 30-60-90 is half an equilateral triangle4 2 90 30
23

39. 30-60-90 is half an equilateral triangle5

40. 30-60-90 is half an equilateral triangle

41. WARM-UP For ABC, if AB = 4, what are BC and AC?
For DEF, if EF = 10, what are GF and GE (height)? (note that DEF is equilateral)
A right triangle has a leg that is 3 and a hypotenuse that is 4, what is the length of the other leg?

42. WARM-UP For ABC, if AB = 4, what are BC and AC? 4 / 2  2 2
For DEF, if EF = 10, what are GF and GE (height)? (note that DEF is equilateral)
A right triangle has a leg that is 3 and a hypotenuse that is 4, what is the length of the other leg?

43. WARM-UP For ABC, if AB = 4, what are BC and AC? 4 / 2  2 2
For DEF, if EF = 10, what are GF and GE (height)? (note that DEF is equilateral) GF=5, GE=5 3
A right triangle has a leg that is 3 and a hypotenuse that is 4, what is the length of the other leg?

44. WARM-UP For ABC, if AB = 4, what are BC and AC? 4 / 2  2 2
For DEF, if EF = 10, what are GF and GE (height)? (note that DEF is equilateral) GF=5, GE=5 3
A right triangle has a leg that is 3 and a hypotenuse that is 4, what is the length of the other leg? 1

45. 30-60-90 is half an equilateral triangle
(Assume short side is opposite small angle)
22 + X2 = 42
X2 = 16 – 4
X =  12
X = 2  3 60 4 2 90 30
23

46. 30-60-90 is half an equilateral triangle10 5 90 30
53

47. 30-60-90 is half an equilateral triangle
1,000,000
500,000 90 30
500,0003

48. 30-60-90 is half an equilateral triangle5 60 90

49. 30-60-90 is half an equilateral triangle
993 30 90 99
198 60

50. 30-60-90 is half an equilateral triangle
43
23 90 30 6

51. 30-60-90 is half an equilateral triangle2 1 90 30 3

52. 30-60-90 is half an equilateral triangle7

53. 30-60-90 is half an equilateral triangle12

54. 45-45-90 is an isosceles triangle
32 3 90 45 3

55. 45-45-90 is an isosceles triangle
1,000,0002
1,000,000 90 45
1,000,000

56. 45-45-90 is an isosceles triangle2 2 2

57. 45-45-90 is an isosceles triangle8 45 90 8
82

58. 45-45-90 is an isosceles triangle
Better 

59. 45-45-90 is an isosceles triangle6 90 45
Better 

60. WARM-UP For ABC, if AC = 52, what are BC and AB?
For GEF, if GF = 6, what are EF and GE? (note that DEF is equilateral)
A right triangle has a leg that is 35 and a hypotenuse that is 7, what is the length of the other leg?

61. WARM-UP For ABC, if AC = 52, what are BC and AB? 5
For GEF, if GF = 6, what are EF and GE? (note that DEF is equilateral)
A right triangle has a leg that is 35 and a hypotenuse that is 7, what is the length of the other leg?

62. WARM-UP For ABC, if AC = 52, what are BC and AB? 5
For GEF, if GF = 6, what are EF and GE? (note:DEF is equilateral) EF = 12, GE 6 3
A right triangle has a leg that is 35 and a hypotenuse that is 7, what is the length of the other leg?

63. WARM-UP For ABC, if AC = 52, what are BC and AB? 5
For GEF, if GF = 6, what are EF and GE? (note:DEF is equilateral) EF = 12, GE 6 3
A right triangle has a leg that is 35 and a hypotenuse that is 7, what is the length of the other leg? 2