Special right triangles

Warm-up

Three quick questions from earlier lessons.

Discover

A square with side 1 is cut along its diagonal. How long is the diagonal?

$\sqrt{2}$
$2$

Key idea

Two triangles whose sides you can know exactly.

45-45-90 (half a square): sides

1 : 1 : \sqrt{2}

. 30-60-90 (half an equilateral triangle): sides

1 : \sqrt{3} : 2

, with the shortest side opposite

30^\circ

and the hypotenuse twice as long.

Worked example

A 30-60-90 triangle with hypotenuse 10.

Short leg

5

Half the hypotenuse.

Long leg

5\sqrt{3}

Short leg × √3.

Your turn, guided

Your turn: a 45-45-90 triangle with hypotenuse $6\sqrt{2}$ . Each leg:

6

Divide by √2.

Practice ★

An isosceles right triangle has legs of length 8. Find the hypotenuse (exact).
An isosceles right triangle has legs of length 6. Find the hypotenuse (exact).

In a 30-60-90 triangle the hypotenuse is 6. Find the side opposite the $30^\circ$ angle.

Watch out!

Practice ★★

In a 30-60-90 triangle the shortest side is 10. Find the longer leg (exact).
In a 30-60-90 triangle the shortest side is 2. Find the longer leg (exact).

Exit ticket

Three questions to check you have got it.

An isosceles right triangle has legs of length 8. Find the hypotenuse (exact).
An isosceles right triangle has legs of length 6. Find the hypotenuse (exact).
An isosceles right triangle has legs of length 11. Find the hypotenuse (exact).

Summary card

Special right triangles

45-45-90: $1 : 1 : \sqrt{2}$ . 30-60-90: $1 : \sqrt{3} : 2$ .

Watch out: √2 vs √3.

End of the lesson

Next lesson: Trigonometric ratios in right triangles Back to the chapter

About this lesson