1. Splash Screen

2. Five-Minute Check (over Lesson 8–1) CCSS Then/Now New Vocabulary
Theorem 8.4: Pythagorean Theorem
Proof: Pythagorean Theorem
Example 1: Find Missing Measures Using the Pythagorean Theorem
Key Concept: Common Pythagorean Triples
Example 2: Use a Pythagorean Triple
Example 3: Standardized Test Example: Use the Pythagorean Theorem
Theorem 8.5: Converse of the Pythagorean Theorem
Theorems: Pythagorean Inequality Theorems
Example 4: Classify Triangles
Lesson Menu

3. Find the geometric mean between 9 and 13.
5-Minute Check 1

4. Find the geometric mean between 9 and 13.
5-Minute Check 1

5. Find the geometric mean between
5-Minute Check 2

6. Find the geometric mean between
5-Minute Check 2

7. Find the altitude a.
A. 4 B.
C. 6 D.
5-Minute Check 3

8. Find the altitude a.
A. 4 B.
C. 6 D.
5-Minute Check 3

9. Find x, y, and z to the nearest tenth.
A. x = 6, y = 8, z = 12
B. x = 7, y = 8.5, z = 15
C. x = 8, y ≈ 8.9, z ≈ 17.9
D. x = 9, y ≈ 10.1, z = 23
5-Minute Check 4

10. Find x, y, and z to the nearest tenth.
A. x = 6, y = 8, z = 12
B. x = 7, y = 8.5, z = 15
C. x = 8, y ≈ 8.9, z ≈ 17.9
D. x = 9, y ≈ 10.1, z = 23
5-Minute Check 4

11. Which is the best estimate for m?
D. 13
5-Minute Check 5

12. Which is the best estimate for m?
D. 13
5-Minute Check 5

13. Mathematical Practices
Content Standards
G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
Mathematical Practices
1 Make sense of problems and persevere in solving them.
4 Model with mathematics.
CCSS

14. You used the Pythagorean Theorem to develop the Distance Formula.
Use the Pythagorean Theorem.
Use the Converse of the Pythagorean Theorem.
Then/Now

15. Pythagorean triple
Vocabulary

16. Concept

17. Concept

18. The side opposite the right angle is the hypotenuse, so c = x.
Find Missing Measures Using the Pythagorean Theorem
A. Find x.
The side opposite the right angle is the hypotenuse, so c = x.
a2 + b2 = c2 Pythagorean Theorem
= c2 a = 4 and b = 7
Example 1

19. Take the positive square root of each side.
Find Missing Measures Using the Pythagorean Theorem
65 = c2 Simplify.
Take the positive square root of each side.
Answer:
Example 1

20. Take the positive square root of each side.
Find Missing Measures Using the Pythagorean Theorem
65 = c2 Simplify.
Take the positive square root of each side.
Answer:
Example 1

21. a2 + b2 = c2 Pythagorean Theorem x2 + 82 = 122 b = 8 and c = 12
Find Missing Measures Using the Pythagorean Theorem
B. Find x.
The hypotenuse is 12, so c = 12.
a2 + b2 = c2 Pythagorean Theorem
x = 122 b = 8 and c = 12
Example 1

22. x2 = 80 Subtract 64 from each side.
Find Missing Measures Using the Pythagorean Theorem
x = 144 Simplify.
x2 = 80 Subtract 64 from each side.
Take the positive square root of each side and simplify.
Answer:
Example 1

23. x2 = 80 Subtract 64 from each side.
Find Missing Measures Using the Pythagorean Theorem
x = 144 Simplify.
x2 = 80 Subtract 64 from each side.
Take the positive square root of each side and simplify.
Answer:
Example 1

24. A. Find x. A. B. C. D.
Example 1A

25. A. Find x. A. B. C. D.
Example 1A

26. B. Find x. A. B. C. D.
Example 1

27. B. Find x. A. B. C. D.
Example 1

28. Concept

29. Use a Pythagorean triple to find x. Explain your reasoning.
Example 2

30. Use a Pythagorean Triple
Notice that 24 and 26 are multiples of 2: 24 = 2 ● 12 and 26 = 2 ● 13. Since 5, 12, 13 is a Pythagorean triple, the missing leg length x is 2 ● 5 or 10.
Answer:
Example 2

31. Check: 242 + 102 = 262 Pythagorean Theorem
Use a Pythagorean Triple
Notice that 24 and 26 are multiples of 2: 24 = 2 ● 12 and 26 = 2 ● 13. Since 5, 12, 13 is a Pythagorean triple, the missing leg length x is 2 ● 5 or 10.
Answer: x = 10
Check: = 262 Pythagorean Theorem ?
676 = 676 Simplify. 
Example 2

32. Use a Pythagorean triple to find x.
B. 15
C. 18
D. 24
Example 2

33. Use a Pythagorean triple to find x.
B. 15
C. 18
D. 24
Example 2

34. Use the Pythagorean Theorem
A 20-foot ladder is placed against a building to reach a window that is 16 feet above the ground. How many feet away from the building is the bottom of the ladder?
A 3
B 4
C 12
D 15
Example 3

35. Use the Pythagorean Theorem
A 20-foot ladder is placed against a building to reach a window that is 16 feet above the ground. How many feet away from the building is the bottom of the ladder?
A 3
B 4
C 12
D 15
Example 3

36. Method 1 Use a Pythagorean triple.
Use the Pythagorean Theorem
Read the Test Item
The distance the ladder is from the house, the height the ladder reaches, and the length of the ladder itself make up the lengths of the sides of a right triangle. You need to find the distance the ladder is from the house, which is a leg of the triangle.
Solve the Test Item
Method 1 Use a Pythagorean triple.
The length of a leg and the hypotenuse are 16 and 20, respectively. Notice that 16 = 4 ● 4 and 20 = 4 ● 5. Since 3, 4, 5 is a Pythagorean triple, the missing length is 4 ● 3 or 12. The answer is C.
Example 3

37. Method 2 Use the Pythagorean Theorem.
Let the distance the ladder is from the house be x.
x = 202 Pythagorean Theorem
x = 400 Simplify.
x2 = 144 Subtract 256 from each side.
x = 12 Take the positive square root of each side.
Answer:
Example 3

38. Method 2 Use the Pythagorean Theorem.
Let the distance the ladder is from the house be x.
x = 202 Pythagorean Theorem
x = 400 Simplify.
x2 = 144 Subtract 256 from each side.
x = 12 Take the positive square root of each side.
Answer: The answer is C.
Example 3

39. A 10-foot ladder is placed against a building
A 10-foot ladder is placed against a building. The base of the ladder is 6 feet from the building. How high does the ladder reach on the building?
A. 6 ft
B. 8 ft
C. 9 ft
D. 10 ft
Example 3

40. A 10-foot ladder is placed against a building
A 10-foot ladder is placed against a building. The base of the ladder is 6 feet from the building. How high does the ladder reach on the building?
A. 6 ft
B. 8 ft
C. 9 ft
D. 10 ft
Example 3

41. Concept

42. Concept

43. The side lengths 9, 12, and 15 can form a triangle.
Classify Triangles
A. Determine whether 9, 12, and 15 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.
Step 1 Determine whether the measures can form a triangle using the Triangle Inequality Theorem.
> 15  > 12  > 9 
The side lengths 9, 12, and 15 can form a triangle.
Example 4

44. Classify Triangles
Step 2 Classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides.
c2 = a2 + b2 Compare c2 and a2 + b2. ?
152 = Substitution ?
225 = 225 Simplify and compare.
Answer:
Example 4

45. Answer: Since c2 = a2 + b2, the triangle is a right triangle.
Classify Triangles
Step 2 Classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides.
c2 = a2 + b2 Compare c2 and a2 + b2. ?
152 = Substitution ?
225 = 225 Simplify and compare.
Answer: Since c2 = a2 + b2, the triangle is a right triangle.
Example 4

46. The side lengths 10, 11, and 13 can form a triangle.
Classify Triangles
B. Determine whether 10, 11, and 13 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.
Step 1 Determine whether the measures can form a triangle using the Triangle Inequality Theorem.
> 13  > 11  > 10 
The side lengths 10, 11, and 13 can form a triangle.
Example 4

47. 169 < 221 Simplify and compare.
Classify Triangles
Step 2 Classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides.
c2 = a2 + b2 Compare c2 and a2 + b2. ?
132 = Substitution ?
169 < 221 Simplify and compare.
Answer:
Example 4

48. 169 < 221 Simplify and compare.
Classify Triangles
Step 2 Classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides.
c2 = a2 + b2 Compare c2 and a2 + b2. ?
132 = Substitution ?
169 < 221 Simplify and compare.
Answer: Since c2 < a2 + b2, the triangle is acute.
Example 4

49. A. Determine whether the set of numbers 7, 8, and 14 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.
A. yes, acute
B. yes, obtuse
C. yes, right
D. not a triangle
Example 4

50. A. Determine whether the set of numbers 7, 8, and 14 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.
A. yes, acute
B. yes, obtuse
C. yes, right
D. not a triangle
Example 4

51. B. Determine whether the set of numbers 14, 18, and 33 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.
A. yes, acute
B. yes, obtuse
C. yes, right
D. not a triangle
Example 4

52. B. Determine whether the set of numbers 14, 18, and 33 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.
A. yes, acute
B. yes, obtuse
C. yes, right
D. not a triangle
Example 4

53. End of the Lesson