1. 7-1 and 7-2: Apply the Pythagorean Theorem
Geometry Chapter 7
7-1 and 7-2: Apply the Pythagorean Theorem

2. Simplifying Radicals
When taking the square root, it can be simplified in one of two ways, depending on the number in the root:
1.) The number is a perfect square
2.) The number is not a perfect square

3. Simplifying Radicals
When taking the square root, it can be simplified in one of two ways, depending on the number in the root:
1.) The number is a perfect square
EX: 𝟑𝟔 𝟔
2.) The number is not a perfect square

4. Simplifying Radicals
When taking the square root, it can be simplified in one of two ways, depending on the number in the root:
1.) The number is a perfect square
EX: 𝟑𝟔 𝟔
2.) The number is not a perfect square
EX: 𝟑𝟐
𝟐×𝟏𝟔
𝟒 𝟐

5. Warm-Up
Simplify the following radicals.
1.) 100
2.) 64
3.) 25
4.) 144

6. Warm-Up
Simplify the following radicals.
5.) 45
6.) 24
7.) 27
8.) 31

7. Apply the Pythagorean Theorem
Objective: Students will be able to find side lengths in right triangles, as well as identify triangles, using the Pythagorean Theorem.
Agenda
Right Triangles
Pythagorean Theorem
Pythagorean Triples
Identify Triangles

8. Right Triangles The sides of a right triangle named as such:
The side opposite the right angle is known as the Hypotenuse
The other two sides are known as the Legs
Hypotenuse
Leg

9. The Pythagorean Theorem
Theorem 7.1 – The Pythagorean Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. 𝒄 𝒃 𝒂
𝑐 2 = 𝑎 2 + 𝑏 2

10. Example 1
Use the Pythagorean Theorem to find the value of x.

11. Example 1 Use the Pythagorean Theorem to find the value of x. 𝑎=5 𝑏=12
𝑐=𝑥

12. Example 1 Use the Pythagorean Theorem to find the value of x. 𝑎=5
𝑏=12
𝑐=𝑥
𝑥 2 =
𝑥 2 =25+144
𝑥 2 =169
𝑥= 169 =13

13. Example 2
Use the Pythagorean Theorem to find the value of x.

14. Example 2 Use the Pythagorean Theorem to find the value of x. 𝑎=𝑥 𝑏=9
𝑐=11

15. Example 2 Use the Pythagorean Theorem to find the value of x.
11 2 = 𝑥
121= 𝑥 2 +81
𝑥 2 =40
𝑥= 40 =2 10
𝑎=𝑥
𝑏=9
𝑐=11

16. Example 3
Use the Pythagorean Theorem to find the value of x. Identify x as either a leg or hypotenuse 𝟓 𝒙 𝟑

17. Example 3
Use the Pythagorean Theorem to find the value of x. Identify x as either a leg or hypotenuse
5 2 = 𝑥
25= 𝑥 2 +9
𝑥 2 =16
𝑥= 16 =𝟒 𝟓 𝒙 𝟑
𝒙=𝟒
Leg

18. Example 4
Use the Pythagorean Theorem to find the value of x. Identify x as either a leg or hypotenuse 𝒙 𝟔 𝟒

19. Example 4
Use the Pythagorean Theorem to find the value of x. Identify x as either a leg or hypotenuse
𝑥 2 =
𝑥 2 =36+16
𝑥 2 =52
𝑥= 52 =𝟐 𝟏𝟑 𝒙 𝟔 𝟒
𝒙=𝟐 𝟏𝟑
Hypotenuse

20. Example 5
Find the Area of the isosceles triangle with side lengths 10 meters, 13 meters, and 13 meters. 𝟏𝟑 𝟏𝟎

21. Example 5 Identify the height by labeling it on the drawing.𝟏𝟑 𝟓 𝒃 𝒉
Recall: Area of a Triangle
𝑨= 𝟏 𝟐 𝒃𝒉

22. Example 5 Use the Pythagorean Theorem to find the value of h.𝟏𝟑 𝟓 𝒃 𝒉
13 2 = ℎ
169=ℎ 2 +25
ℎ 2 =144
ℎ= 144 =12

23. Example 5
Solve for area 𝟏𝟑 𝟓 𝟏𝟎 𝟏𝟐
𝐴= 1 2 𝑏ℎ
𝐴= 1 2 (10)(12)
𝑨=𝟔𝟎

24. Example 6
Find the area of the given triangle. 𝟏𝟖 𝟑𝟎

25. Example 6 Find the area of the given triangle. 𝟑𝟎 18 2 = ℎ 2 + 15 2
324=ℎ
ℎ 2 =99
ℎ= 99 =3 11 𝟏𝟖 𝟏𝟓 𝒉

26. Example 6 Find the area of the given triangle. 𝟑𝟎 𝐴= 1 2 𝑏ℎ 𝟏𝟓
𝐴= 1 2 (30)(3 11 )
𝑨=𝟒𝟓 𝟏𝟏 𝟏𝟖 𝟏𝟓 𝒉

27. Example 6
Find the area of the given triangle. 𝟐𝟔 𝟐𝟎

28. Example 6 Find the area of the given triangle. 𝟐𝟔 𝟏𝟎 26 2 = ℎ 2 + 10 2
676=ℎ
ℎ 2 =576
ℎ= 576 =24 𝟐𝟎 𝒉 𝟏𝟎

29. Example 6 Find the area of the given triangle. 𝟐𝟔 𝟏𝟎 𝐴= 1 2 𝑏ℎ
𝐴= 1 2 (20)(24)
𝑨=𝟐𝟒𝟎 𝟐𝟎 𝟐𝟒 𝟏𝟎

30. Pythagorean Triples
When a right triangle has side lengths that are all whole numbers, we call them a Pythagorean Triple.
Examples of Pythagorean Triples:
3 – 4 – 5
5 – 12 – 13
8 – 15 – 17
7 – 24 – 25

31. Pythagorean Triples
When a right triangle has side lengths that are all whole numbers, we call them a Pythagorean Triple.
Examples of Pythagorean Triples:
3 – 4 – 5
6 – 8 – 10
9 – 12 – 15
15 – 20 – 25
12 – 16 – 20
5 – 12 – 13
8 – 15 – 17
7 – 24 – 25
10 – 24 – 26
16 – 30 – 34
14 – 48 – 50
15 – 36 – 39
20 – 48 – 52
24 – 45 – 51

32. Example 7
Find the value of x on both triangles 𝟐𝟓 𝟕 𝒙 𝒙 𝟖 𝟔

33. Example 7
Find the value of x on both triangles 𝟐𝟓 𝟕 𝒙 𝒙 𝟖 𝟔
𝒙=𝟏𝟎
𝒙=𝟐𝟒

34. Example 7
Find the value of x on both triangles 𝟏𝟓 𝒙 𝟏𝟐 𝒙 𝟓 𝟒

35. Example 7
Find the value of x on both triangles 𝟏𝟓 𝒙 𝟏𝟐
𝒙=𝟗 𝒙 𝟓 𝟒

36. Example 7 Find the value of x on both triangles 𝒙=𝟗 𝑥 2 = 4 2 + 5 2𝟏𝟓 𝒙 𝟏𝟐
𝒙=𝟗
𝑥 2 =
𝑥 2 =16+25
𝑥 2 =41
𝒙= 𝟒𝟏 ≈𝟔.𝟒𝟎𝟑 𝒙 𝟓 𝟒

37. The Pythagorean Theorem
Theorem 7.2 – Converse of the Pythagorean Theorem: If the square of the length on the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
If 𝑐 2 = 𝑎 2 + 𝑏 2 ,
Then ∆𝐴𝐵𝐶 is a right triangle 𝒄 𝒃 𝒂 𝑨 𝑩 𝑪

38. Example 8 Tell whether the given triangle is a right triangle. 𝟗 𝟑 𝟑𝟒𝟏𝟓
𝟑 𝟑𝟒 𝟗

39. Example 8 Tell whether the given triangle is a right triangle.𝟏𝟓
𝟑 𝟑𝟒 𝟗
(3 34 ) 2 =
9×34=225+81
𝟑𝟎𝟔=𝟑𝟎𝟔

40. Example 8 Tell whether the given triangle is a right triangle.𝟏𝟓
𝟑 𝟑𝟒 𝟗
(3 34 ) 2 =
9×34=225+81
𝟑𝟎𝟔=𝟑𝟎𝟔
Thus, the triangle is a Right Triangle

41. Example 9 Tell whether the given triangle is a right triangle. 𝟏𝟒 𝟐𝟐𝟐𝟔 𝟐𝟐 𝟏𝟒

42. Example 9 Tell whether the given triangle is a right triangle.
26 2 =
676=
𝟔𝟕𝟔≠𝟔𝟖𝟎 𝟐𝟔 𝟐𝟐 𝟏𝟒
Thus, the triangle is NOT a Right Triangle

43. The Pythagorean Theorem
Theorem 7.3: If the square of the length on the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is an acute triangle. 𝒄 𝒃 𝒂 𝑨 𝑩 𝑪
If 𝑐 2 < 𝑎 2 + 𝑏 2 ,
Then ∆𝐴𝐵𝐶 is an acute triangle

44. The Pythagorean Theorem
Theorem 7.4: If the square of the length on the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is an obtuse triangle. 𝒄 𝒃 𝒂 𝑨 𝑩 𝑪
If 𝑐 2 > 𝑎 2 + 𝑏 2 ,
Then ∆𝐴𝐵𝐶 is an obtuse triangle

45. Example 10
Determine if a triangle with side lengths 4.3, 5.2, and 6.1 form a right, acute, or obtuse triangle.

46. Example 10
Determine if a triangle with side lengths 4.3, 5.2, and 6.1 form a right, acute, or obtuse triangle.
𝑎=4.2
𝑏=5.2
𝑐=6.1
6.1 2 =
37.21=
𝟑𝟕.𝟐𝟏<𝟒𝟓.𝟓𝟑

47. Example 10
Determine if a triangle with side lengths 4.3, 5.2, and 6.1 form a right, acute, or obtuse triangle.
𝑎=4.2
𝑏=5.2
𝑐=6.1
6.1 2 =
37.21=
𝟑𝟕.𝟐𝟏<𝟒𝟓.𝟓𝟑
Thus, the triangle is an Acute Triangle.

48. Example 10 Side Lengths: 4, 4 3 , 8 Side Lengths: 11, 20, 23
Determine if the triangles with the given side lengths form a right, acute, or obtuse triangle.
Side Lengths: 4, 4 3 , 8
Side Lengths: 11, 20, 23

49. Example 10 Side Lengths: 4, 4 3 , 8 Side Lengths: 11, 20, 23
Determine if the triangles with the given side lengths form a right, acute, or obtuse triangle.
Side Lengths: 4, 4 3 , 8
Side Lengths: 11, 20, 23
8 2 = (4 3 )
64=(16×3)+16
𝟔𝟒=𝟔𝟒
Thus, the triangle is a Right Triangle.

50. Example 10 Side Lengths: 4, 4 3 , 8 Side Lengths: 11, 20, 23
Determine if the triangles with the given side lengths form a right, acute, or obtuse triangle.
Side Lengths: 4, 4 3 , 8
Side Lengths: 11, 20, 23
8 2 = (4 3 )
64=(16×3)+16
𝟔𝟒=𝟔𝟒
23 2 =
529=
𝟓𝟐𝟗>𝟓𝟐𝟏
Thus, the triangle is a Right Triangle.
Thus, the triangle is an Obtuse Triangle.

51. 𝟐𝟔 𝟏𝟎 𝒙 𝒙 𝟏𝟐 𝟏𝟔 𝟖 𝒙 𝟔

52. 𝟑𝟗 𝟏𝟓 𝒙 𝒙 𝟓 𝟏𝟑 𝒙 𝟒𝟎 𝟑𝟎