Lesson 9

The Converse

Lesson Narrative

This lesson guides students through a proof of the converse of the Pythagorean Theorem. Then students have an opportunity to decide if a triangle with three given side lengths is or is not a right triangle.

Learning Goals

Teacher Facing

  • Determine whether a triangle with given side lengths is a right triangle using the converse of the Pythagorean Theorem.
  • Generalize (orally) that if the side lengths of a triangle satisfy the equation $a^2+b^2=c^2$ then the triangle must be a right triangle.
  • Justify (orally) that a triangle with side lengths 3, 4, and 5 must be a right triangle.

Student Facing

Let’s figure out if a triangle is a right triangle.

Learning Targets

Student Facing

  • I can explain why it is true that if the side lengths of a triangle satisfy the equation $a^2+b^2=c^2$ then it must be a right triangle.
  • If I know the side lengths of a triangle, I can determine if it is a right triangle or not.

CCSS Standards

Addressing

Building Towards

Glossary Entries

  • Pythagorean Theorem

    The Pythagorean Theorem describes the relationship between the side lengths of right triangles.

    The diagram shows a right triangle with squares built on each side.  If we add the areas of the two small squares, we get the area of the larger square.

    The square of the hypotenuse is equal to the sum of the squares of the legs. This is written as \(a^2+b^2=c^2\).

    a right triangle with squares built on each side
  • hypotenuse

    The hypotenuse is the side of a right triangle that is opposite the right angle. It is the longest side of a right triangle.

    Here are some right triangles. Each hypotenuse is labeled.

    Four right triangles of different sizes and orientations each with two legs and a hypotenuse opposite the right angle.
  • legs

    The legs of a right triangle are the sides that make the right angle. 

    Here are some right triangles. Each leg is labeled.

    Four right triangles of different sizes and orientations each with two legs and a hypotenuse opposite the right angle.