The Pythagorean theorem

Warm-up

Three quick questions from earlier lessons.

Discover

A right triangle has legs 3 and 4. Squares are drawn on all three sides. The squares on the legs have areas 9 and 16. What is the area of the square on the hypotenuse?

Set the legs to 3 and 4, then try other lengths. Which rule does the hypotenuse follow?

The hypotenuse is √(a² + b²)
The hypotenuse is a + b

Key idea

The Pythagorean theorem.

In a right triangle with legs

a

b

and hypotenuse

c

(the side opposite the right angle, the longest side):

a^2 + b^2 = c^2

. To find a leg, subtract:

b^2 = c^2 - a^2

Worked example

Hypotenuse 10, one leg 6.

b^2 = 100 - 36 = 64

Subtract the squares.

b = 8

Square root.

Your turn, guided

Your turn: legs 8 and 15. The hypotenuse.

17

√(64 + 225) = √289.

Practice ★

A right triangle has legs 10 and 24. How long is the hypotenuse?
A right triangle has legs 20 and 48. How long is the hypotenuse?

A right triangle has legs 2 and 4. Find the exact length of the hypotenuse.

Watch out!

Practice ★★

A right triangle has hypotenuse 15 and one leg 9. How long is the other leg?
A right triangle has hypotenuse 25 and one leg 15. How long is the other leg?

Exit ticket

Three questions to check you have got it.

A right triangle has legs 10 and 24. How long is the hypotenuse?
A right triangle has legs 20 and 48. How long is the hypotenuse?
A right triangle has legs 15 and 36. How long is the hypotenuse?

Summary card

Pythagorean theorem

$a^2 + b^2 = c^2$ , with $c$ the hypotenuse.

Watch out: for a leg, subtract the squares.

End of the lesson

Next lesson: The converse of the Pythagorean theorem; applications and distances Back to the chapter

About this lesson