Pythagorean Theorem & Its Converse Skill 40

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

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 .

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.

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.

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

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

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

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

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.

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.

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.

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.

#40: Pythagorean Theorem and Its Converse Questions? Summarize Notes Homework Quiz