# 11.4: The Pythagorean Theorem and Its Converse

**At Grade**Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

- Use the Pythagorean theorem.
- Use the converse of the Pythagorean theorem.
- Solve real-world problems using the Pythagorean theorem and its converse.

## Vocabulary

Terms introduced in this lesson:

- Pythagorean theorem
- hypotenuse
- legs
- converse

## Teaching Strategies and Tips

Students learn that the Pythagorean theorem relates the lengths of the sides of a right triangle to each other.

Students also learn that the Pythagorean theorem has a converse.

- It can be used to verify that a triangle is a right triangle.
- If it can be shown that the three sides of a triangle make the equation \begin{align*}a^2 + b^2 = c^2\end{align*} true, then the triangle is a right triangle.

Remind students of the following few prerequisites from geometry:

- A right triangle is one that contains a \begin{align*}90\;\mathrm{degree}\end{align*} angle.
- The side of the triangle opposite the \begin{align*}90\;\mathrm{degree}\end{align*} angle is called the hypotenuse.
- The sides of the triangle adjacent to the \begin{align*}90\;\mathrm{degree}\end{align*} angle are called the legs.
- The longest side of a right triangle is the hypotenuse.

Encourage students to say the Pythagorean theorem in words.

Point out that knowing the value of two variables in the Pythagorean theorem is sufficient for determining the third. See Example 4.

Suggest that students discard negative solutions obtained from radical equations in this lesson since the lengths of sides of triangles are nonnegative. See Example 5.

Specify how many decimal places are required of students when rounding.

## Error Troubleshooting

General Tip: Remind students to set the value of \begin{align*}c\end{align*} as the length of the hypotenuse; the values for \begin{align*}a\end{align*} and \begin{align*}b\end{align*} can be switched.

In *Review Question* 18, assume that the hypotenuse is also the diameter of the circle.

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