# The Converse of the Pythagorean Theorem

Videos to help Grade 8 students explain a proof of the converse of the Pythagorean Theorem.and apply the theorem and its converse to solve problems.

New York State Common Core Math Grade 8, Module 7, Lesson 16

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Lesson 16 Student Outcomes

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

Lesson 16 Summary

The converse of the Pythagorean Theorem states that if a triangle with side lengths a, b, and c satisfies a2 + b2 = c2, then the triangle is a right triangle.
The converse can be proven using concepts related to congruence.

Lesson 16 Classwork

Discussion
So far you have seen three different proofs of the Pythagorean Theorem: If the lengths of the legs of a right triangle are a and b, and the length of the hypotenuse is c, then a2 + b2 = c2.

The converse of the Pythagorean Theorem:
If the lengths of three sides of a triangle, a, b, and c satisfy c2 = a2 + b2 then the triangle is a right triangle, and furthermore, the side of length c is opposite the right angle.

Exercises
1. Is the triangle with leg lengths of 3 mi., 8 mi., and hypotenuse of length $$\sqrt {73}$$ mi. a right triangle? Show your work, and answer in a complete sentence.
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.
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.
4. Is the triangle with leg lengths of 9 in., 9 in., and hypotenuse of length $$\sqrt {175}$$ in. a right triangle? Show your work, and answer in a complete sentence.
5. Is the triangle with leg lengths of $$\sqrt {28}$$ cm, 6 cm, and hypotenuse of length 8 cm a right triangle? Show your work, and answer in a complete sentence.
6. What is the length of the unknown side of the right triangle shown below? Show your work, and answer in a complete sentence.
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.

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