Lesson 9The Converse

Learning Goal

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

Learning Targets

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.

Lesson Terms

hypotenuse
legs
Pythagorean Theorem

Warm Up: The Hands of a Clock

Consider the tips of the hands of an analog clock that has an hour hand that is 3 centimeters long and a minute hand that is 4 centimeters long.

Over the course of a day:

The image of a circle that represent an analog clock. On the circle are 12 evenly spaced tick marks. There are two hands on the clock. One hand is labeled 3, begins in the center of the circle and extends upward and to the right, and points to the third tick mark from the top. The other hand is labeled 4, begins in the center of the circle and extends upward and to the left. It points to the eleventh tick mark from the top.

What is the farthest apart the two tips get?
What is the closest the two tips get?
Are the two tips ever exactly five centimeters apart?

Activity 1: Proving the Converse

Here are three triangles with two side lengths measuring 3 and 4 units, and the third side of unknown length.

A figure of three triangles each with 2 given side lengths and one unknown side length. The first triangle has a horizontal side of 4, a side length slanted upward and to the left of 3, and the third side length labeled x. The middle triangle has a horizontal side length of 4, a second side length slanted upward and to the right of 3, and the third side length labeled y. The third triangle is a right triangle with a horizontal side length of 4, a vertical side length of 3, and the third side is labeled z.

Sort the following six numbers from smallest to largest. Put an equal sign between any you know to be equal. Be ready to explain your reasoning.

$1 5 7 x y z$

Are you ready for more?

A related argument also lets us distinguish acute from obtuse triangles using only their side lengths.

Decide if triangles with the following side lengths are acute, right, or obtuse. In right or obtuse triangles, identify which side length is opposite the right or obtuse angle.

$x = 15$ , $y = 20$ , $z = 8$
$x = 8$ , $y = 15$ , $z = 13$
$x = 17$ , $y = 8$ , $z = 15$

Activity 2: Calculating Legs of Right Triangles

Problem 1

Given the information provided for the right triangles shown here, find the unknown leg lengths to the nearest tenth.

Problem 2

The triangle shown here is not a right triangle. What are two different ways you change one of the values so it would be a right triangle? Sketch these new right triangles, and clearly label the right angle.

A triangle with side lengths of 4, 6, and 7.

Lesson Summary

What if it isn’t clear whether a triangle is a right triangle or not? Here is a triangle:

A triangle with side lengths labeled 15, 17 and 8.

Is it a right triangle? It’s hard to tell just by looking, and it may be that the sides aren’t drawn to scale.

If we have a triangle with side lengths $a$ , $b$ , and $c$ , with $c$ being the longest of the three, then the converse of the Pythagorean Theorem tells us that any time we have $a^{2} + b^{2} = c^{2}$ , we must have a right triangle. Since $8^{2} + 15^{2} = 64 + 225 = 289 = 17^{2}$ , any triangle with side lengths 8, 15, and 17 must be a right triangle.

Together, the Pythagorean Theorem and its converse provide a one-step test for checking to see if a triangle is a right triangle just using its side lengths. If $a^{2} + b^{2} = c^{2}$ , it is a right triangle. If $a^{2} + b^{2} \neq c^{2}$ , it is not a right triangle.