Special Right Triangles on the SAT

Why the SAT Emphasizes Special Right Triangles

Special right triangles are time savers. When you see a $45^\circ$ angle or a $30^\circ$ angle, you do not need to run the full Pythagorean Theorem. You can use fixed ratios and move on.

This lesson lays out the two key triangles, shows how to scale them, and helps you keep the side labels straight. Knowing these ratios cold can turn a long problem into a quick one.

A Simple Definition Unlocks Special Right Triangles

A $45$-$45$-$90$ triangle has equal legs and a hypotenuse that is $\sqrt{2}$ times a leg. The ratio is $1 : 1 : \sqrt{2}$.

A $30$-$60$-$90$ triangle has a short leg, a long leg, and a hypotenuse in the ratio $1 : \sqrt{3} : 2$. The short leg is always opposite the $30^\circ$ angle.

Work Through Special Right Triangles Step by Step

Use the special triangle ratio to find the missing side without extra algebra.

Use Desmos to Check Special Right Triangles

Desmos can compute the numeric value of $\sqrt{2}$ or $\sqrt{3}$ quickly if you need a decimal.

Algebra is faster for exact form. Desmos is useful when the SAT asks for approximations.

Desmos is faster only if you need a decimal check, but memorized ratios are faster for exact answers.

Expert move: Use Desmos for decimal checks only if an approximation is required. Exact $45$-$45$-$90$ and $30$-$60$-$90$ ratios are faster and usually match the answer choices.

When to skip Desmos: If the answer choices are radicals, use the memorized ratios instead of decimals.

• Desmos features used: numeric evaluation.
• Common mistake: mixing the $45-45-90$ ratio with the $30-60-90$ ratio.

Practice Special Right Triangles with SAT-Style Questions

Use special right triangle ratios.

Key Takeaways to Remember for Special Right Triangles

• $45-45-90$: legs equal, hypotenuse is $\text{leg} \cdot \sqrt{2}$.
• $30-60-90$: sides are $1 : \sqrt{3} : 2$.
• Desmos is useful for decimal approximations.