1. Pythagorean Theorem & Its Converse
Skill 40

2. Objective
HSG-SRT.8: Students are responsible for using and applying the Pythagorean Theorem and its Converse.

3. Definitions
A Pythagorean triple is a set of nonzero whole numbers a, b, and c that satisfy the equation 𝑎 2 + 𝑏 2 = 𝑐 2 .

4. Theorem 71: Pythagorean Theorem
The square of the hypotenuse is equal to the sum of the squares of the other two sides.
Theorem 72: Converse of the Pythagorean Theorem
For any three positive numbers; a, b, and c such that 𝑎 2 + 𝑏 2 = 𝑐 2 , there exists a triangle with sides a, b, and c, and ever such triangle has a right angle between the sides of lengths a and b.

5. Theorem 73: Acute Triangle Theorem
If 𝑎 2 + 𝑏 2 > 𝑐 2 , then it is an acute triangle.
Theorem 74: Obtuse Triangle Theorem
If 𝑎 2 + 𝑏 2 < 𝑐 2 , then it is an obtuse triangle.

6. Example 1; Finding the Length of the Hypotenuse
a) Find the value of x. 20 21 x
𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐
𝟐 𝟏 𝟐 +𝟐 𝟎 𝟐 = 𝒙 𝟐
𝟒𝟒𝟏+𝟒𝟎𝟎= 𝒙 𝟐
𝟖𝟒𝟏= 𝒙 𝟐
𝒙=𝟐𝟗

7. Example 1; Finding the Length of the Hypotenuse
b) Find the value of y.
𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐 24 10 y
𝟏𝟎 𝟐 + 𝟐𝟒 𝟐 = 𝒚 𝟐
𝟏𝟎𝟎+𝟓𝟕𝟒= 𝒚 𝟐
𝟓𝟕𝟒= 𝒚 𝟐
𝒚=𝟐𝟓.𝟗𝟔

8. Example 2; Finding the Length of a Leg
a) Find the value of x.
𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐 20 8 x
𝟖 𝟐 + 𝒙 𝟐 = 𝟐𝟎 𝟐
𝟔𝟒+ 𝒙 𝟐 =𝟒𝟎𝟎
𝒙 𝟐 =𝟑𝟑𝟔
𝒙=𝟏𝟖.𝟑𝟑

9. Example 2; Finding the Length of a Leg
b) Find the value of y. 6 12 y
𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐
𝟔 𝟐 + 𝒚 𝟐 = 𝟏𝟐 𝟐
𝟑𝟔+ 𝒚 𝟐 =𝟏𝟒𝟒
𝒚 𝟐 =𝟏𝟎𝟖
𝒚=𝟏𝟎.𝟑𝟗

10. Example 3; Finding Distance
a) The size of a computer monitor is the length of its diagonal. You want to buy a 19-in. monitor that has a height of 11 in. What is the width of the monitor? Round to the nearest tenth of an inch. 11 19 y
𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐
𝟏𝟏 𝟐 + 𝒚 𝟐 = 𝟏𝟗 𝟐
𝟏𝟐𝟏+ 𝒚 𝟐 =𝟑𝟔𝟏
𝒚 𝟐 =𝟐𝟒𝟎
𝒚=𝟏𝟓.𝟒𝟗
The monitor is 15 inches wide.

11. Example 3; Finding Distance
b) James leans a 12-ft ladder against the side of a house. The base of the ladder is 4-ft from the house. To the nearest tenth of a foot, how high on the house does the ladder reach? 12 4 h
𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐
𝟒 𝟐 + 𝒉 𝟐 = 𝟏𝟐 𝟐
𝟏𝟔+ 𝒉 𝟐 =𝟏𝟒𝟒
𝒚 𝟐 =𝟏𝟐𝟖
𝒚=𝟏𝟏.𝟑
The ladder will reach 11.3 feet high.

12. Example 4; Classify the Triangle
a) A triangle has side lengths of 6, 11, and 14.
𝒂 𝟐 + 𝒃 𝟐 ∎ 𝒄 𝟐 6 14 11
𝟔 𝟐 + 𝟏𝟏 𝟐 ∎ 𝟏𝟒 𝟐
𝟑𝟔+𝟏𝟐𝟏 ∎ 𝟏𝟗𝟔
𝟏𝟓𝟕 ∎ 𝟏𝟗𝟔
𝟏𝟓𝟕<𝟏𝟗𝟔
𝒂 𝟐 + 𝒃 𝟐 < 𝒄 𝟐
The triangle is obtuse.

13. Example 4; Classify the Triangle
b) A triangle has side lengths of 7, 8, and 9.
𝒂 𝟐 + 𝒃 𝟐 ∎ 𝒄 𝟐 7 9 8
𝟕 𝟐 + 𝟖 𝟐 ∎ 𝟗 𝟐
𝟒𝟗+𝟔𝟒 ∎ 𝟖𝟏
𝟏𝟏𝟑 ∎ 𝟖𝟏
𝟏𝟏𝟑>𝟖𝟏
𝒂 𝟐 + 𝒃 𝟐 > 𝒄 𝟐
The triangle is acute.

14. #40: Pythagorean Theorem and Its Converse
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