Activating Prior Knowledge M7:LSN16 The Converse of the Pythagorean Theorem Activating Prior Knowledge Are the triangles shown below right triangles? Prove yes or no using the Pythagorean Theorem. Tie to LO

Learning Objective Students explain a proof of the converse of the Pythagorean Theorem. Students apply the theorem and its converse to solve problems. CFU

M7:LSN16 The Converse of the Pythagorean Theorem Concept Development What do you recall about the meaning of the word converse? Be prepared to provide an example. The converse is when the hypothesis and conclusion of a theorem are reversed. Example 1: If it is a right angle, then the angle measure is 90˚. Converse: If the angle measure is 90˚, then it is a right angle. Example 2: If it is raining, I will study inside the house. Converse: If I study inside the house, it is raining. CFU

Converse of the Pythagorean Theorem M7:LSN16 The Converse of the Pythagorean Theorem Concept Development Converse of the Pythagorean Theorem What do we know or not know about each of these triangles? In the first triangle, 𝐴𝐵𝐶, we know that 𝑎 2 + 𝑏 2 = 𝑐 2 . We do not know if angle 𝐶 is a right angle. In the second triangle, 𝐴′𝐵′𝐶′, we know that it is a right triangle. What conclusions can we draw from this? CFU

Converse of the Pythagorean Theorem M7:LSN16 The Converse of the Pythagorean Theorem Concept Development Converse of the Pythagorean Theorem In the first triangle, 𝐴𝐵𝐶, we know that 𝑎 2 + 𝑏 2 = 𝑐 2 . We do not know if angle 𝐶 is a right angle. In the second triangle, 𝐴′𝐵′𝐶′, we know that it is a right triangle. What conclusions can we draw from this? By applying the Pythagorean Theorem to △𝐴′𝐵′𝐶′, we get 𝐴′𝐵′ 2 = 𝑎 2 + 𝑏 2 . Since we are given 𝑐 2 = 𝑎 2 + 𝑏 2 , then by substitution, 𝐴′𝐵′ 2 = 𝑐 2 , and then 𝐴′𝐵′ =𝑐. Since 𝑐 is also 𝐴𝐵 , then 𝐴′𝐶′ = 𝐴𝐶 . That means that both triangles have sides, 𝑎, 𝑏, and 𝑐, that are the exact same lengths. Therefore, one triangle would map onto the other triangle showing a congruence. Congruence is degree preserving, which means that ∠𝐴𝐶𝐵 is a right angle, i.e., 90˚=∠ 𝐴 ′ 𝐶 ′ 𝐵 ′ =∠𝐴𝐶𝐵. CFU

Concept Development/Guided Practice M7:LSN16 The Converse of the Pythagorean Theorem Concept Development/Guided Practice Exercise 1 Is the triangle with leg lengths of 𝟑 mi., 𝟖 mi., and hypotenuse of length 𝟕𝟑 mi. a right triangle? Show your work, and answer in a complete sentence. 𝟑 𝟐 + 𝟖 𝟐 = 𝟕𝟑 𝟐 𝟗+𝟔𝟒=𝟕𝟑 𝟕𝟑=𝟕𝟑 Yes, the triangle with leg lengths of 𝟑 mi., 𝟖 mi., and hypotenuse of length 𝟕𝟑 mi. is a right triangle because it satisfies the Pythagorean Theorem. CFU

Concept Development/Guided Practice M7:LSN16 The Converse of the Pythagorean Theorem Concept Development/Guided Practice Exercise 2 What is the length of the unknown side of the right triangle shown below? Show your work, and answer in a complete sentence. Provide an exact answer and an approximate answer rounded to the tenths place. Let 𝒄 represent the hypotenuse of the triangle. 𝟏 𝟐 + 𝟒 𝟐 = 𝒄 𝟐 𝟏+𝟏𝟔= 𝒄 𝟐 𝟏𝟕= 𝒄 𝟐 𝟏𝟕 =𝒄 𝟒.𝟏≈𝒄 The length of the hypotenuse of the right triangle is exactly 𝟏𝟕 inches and approximately 𝟒.𝟏 inches. CFU

Time to work with your partners! You have 15 minutes! M7:LSN16 The Converse of the Pythagorean Theorem Concept Development/Guided Practice Exercises 3-7 Time to work with your partners! You have 15 minutes! 15 minutes CFU

Concept Development/Guided Practice M7:LSN16 The Converse of the Pythagorean Theorem Concept Development/Guided Practice Exercise 3 What is the length of the unknown side of the right triangle shown below? Show your work, and answer in a complete sentence. Provide an exact answer and an approximate answer rounded to the tenths place. Let 𝒄 represent the hypotenuse of the triangle. 𝟐 𝟐 + 𝟔 𝟐 = 𝒄 𝟐 𝟒+𝟑𝟔= 𝒄 𝟐 𝟒𝟎= 𝒄 𝟐 𝟒𝟎 =𝒄 𝟐 𝟑 × 𝟓 =𝒄 𝟐 𝟐 × 𝟐 × 𝟓 =𝒄 𝟐 𝟏𝟎 =𝒄 The length of the hypotenuse of the right triangle is exactly 𝟐 𝟏𝟎 mm and approximately 𝟔.𝟑 mm. CFU

Concept Development/Guided Practice M7:LSN16 The Converse of the Pythagorean Theorem Concept Development/Guided Practice Exercise 4 Is the triangle with leg lengths of 𝟗 in., 𝟗 in., and hypotenuse of length 𝟏𝟕𝟓 in. a right triangle? Show your work, and answer in a complete sentence. 𝟗 𝟐 + 𝟗 𝟐 = 𝟏𝟕𝟓 𝟐 𝟖𝟏+𝟖𝟏=𝟏𝟕𝟓 𝟏𝟔𝟐≠𝟏𝟕𝟓 No, the triangle with leg lengths of 𝟗 in., 𝟗 in., and hypotenuse of length 𝟏𝟕𝟓 in. is not a right triangle because the lengths do not satisfy the Pythagorean Theorem. CFU

Concept Development/Guided Practice M7:LSN16 The Converse of the Pythagorean Theorem Concept Development/Guided Practice Exercise 5 Is the triangle with leg lengths of 𝟐𝟖 cm, 𝟔 cm, and hypotenuse of length 𝟖 cm a right triangle? Show your work, and answer in a complete sentence. 𝟐𝟖 𝟐 + 𝟔 𝟐 = 𝟖 𝟐 𝟐𝟖+𝟑𝟔=𝟔𝟒 𝟔𝟒=𝟔𝟒 Yes, the triangle with leg lengths of 𝟐𝟖 cm, 𝟔 cm, and hypotenuse of length 𝟖 cm is a right triangle because the lengths satisfy the Pythagorean Theorem. CFU

Concept Development/Guided Practice M7:LSN16 The Converse of the Pythagorean Theorem Concept Development/Guided Practice Exercise 6 What is the length of the unknown side of the right triangle shown below? Show your work, and answer in a complete sentence. Let 𝒄 represent the hypotenuse of the triangle. 𝟑 𝟐 + 𝟐𝟕 𝟐 = 𝒄 𝟐 𝟗+𝟐𝟕= 𝒄 𝟐 𝟑𝟔= 𝒄 𝟐 𝟑𝟔 = 𝒄 𝟐 𝟔=𝒄 The length of the hypotenuse of the right triangle is 𝟔 ft. CFU

Concept Development/Guided Practice M7:LSN16 The Converse of the Pythagorean Theorem Concept Development/Guided Practice Exercise 7 The triangle shown below is an isosceles right triangle. Determine the length of the legs of the triangle. Show your work, and answer in a complete sentence. Let 𝒙 represent the length of the side of the isosceles triangle. 𝒙 𝟐 + 𝒙 𝟐 = 𝟏𝟖 𝟐 𝟐 𝒙 𝟐 =𝟏𝟖 𝟐 𝒙 𝟐 𝟐 = 𝟏𝟖 𝟐 𝒙 𝟐 =𝟗 𝒙 𝟐 = 𝟗 𝒙=𝟑 The leg lengths of the isosceles triangle are 𝟑 cm. CFU

M7:LSN16 The Converse of the Pythagorean Theorem LESSON SUMMARY The converse of the Pythagorean Theorem states that if a triangle with side lengths 𝒂, 𝒃, and 𝒄 satisfies 𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐 , then the triangle is a right triangle. The converse can be proven using concepts related to congruence. CFU

Closure Homework – Module 7 page 81 Problem Set #1-5 CFU What did we learn today? If the side lengths of a triangle do not satisfy a2+b2=c2, then what do we know about the triangle? 3. What does the converse of the Pythagorean Theorem state? Homework – Module 7 page 81 Problem Set #1-5 CFU