Use properties of special right triangles (30–60–90, 45–45–90). | Right triangles and trigonometry | SAT Math

Use 45-45-90 and 30-60-90 ratios to get exact side lengths fast.

Core Idea

In a 45-45-90 triangle, the sides are in the ratio 1:1:sqrt(2). In a 30-60-90 triangle, the sides are in the ratio 1:sqrt(3):2, with 1 opposite 30 degrees and 2 as the hypotenuse.

Understanding

These two triangles appear constantly on the SAT because they give exact side lengths without a calculator. The key is knowing which side goes where.

45-45-90: two equal legs, hypotenuse = leg × $\sqrt 2$ . Think of it as half a square cut along its diagonal.

30-60-90: the short leg (opposite 30°) is half the hypotenuse. The long leg (opposite 60°) is short leg × $\sqrt 3$ . Think of it as half an equilateral triangle.

When you see a problem with these angle pairs, skip the Pythagorean theorem — the ratios are faster and cleaner.

Step by Step

Identify the angle pair: 45-45-90 or 30-60-90.
Determine which side you're given and which position it occupies in the ratio.
Scale the entire ratio to match the given side.
Read off the unknown side.

Misconceptions

Mixing up $\sqrt 2$ and $\sqrt 3$ — $\sqrt 2$ goes with 45-45-90, $\sqrt 3$ goes with 30-60-90.
In 30-60-90: putting $\sqrt 3$ on the hypotenuse instead of the long leg.
In 30-60-90: confusing which leg is opposite 30° and which is opposite 60°.

Question

Worked Example

In a 30-60-90 triangle, the side opposite the 30° angle has length 5. What is the length of the side opposite the 60° angle?

Select an answer to see the explanation