right triangles & trig

Right triangles and basic trigonometry show up again and again on the SAT. Master the Pythagorean Theorem, special triangles (45-45-90 and 30-60-90), the meaning of sine, cosine, and tangent, and how to use similarity to solve for missing lengths and angles. This page will show you exactly what you need, step by step.

what is a right triangle?

A right triangle has one 90° angle. The side opposite the right angle is called the hypotenuse. The other two sides are called legs. Right triangles follow the Pythagorean Theorem:

a^2 + b^2 = c^2

Where $c$ is the hypotenuse and $a, b$ are the legs.

special right triangles

The SAT especially loves two types: Isosceles right triangle (45°-45°-90°) and 30°-60°-90° triangles.

45°-45°-90° Triangle:

\text{legs } = x,\ \text{hypotenuse } = x\sqrt{2}

Both legs are equal. Hypotenuse is $\\sqrt{2}$ times a leg.

30°-60°-90° Triangle:

\text{short leg} = x,\ \text{long leg} = x\sqrt{3},\ \text{hypotenuse} = 2x

Opposite 30°: shortest leg; opposite 60°: longer leg.

For example:

If the hypotenuse of a 45-45-90 triangle is 10, each leg is $\\frac{10}{\\sqrt{2}} = 5\\sqrt{2}$ .

trigonometric ratios

For any right triangle, you can use SOHCAHTOA:

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}

\tan\theta = \frac{\text{opposite}}{\text{adjacent}}

For example:

In a right triangle, if one leg is 3, the other is 4, and the hypotenuse is 5:

\sin\theta = \frac{3}{5},\ \cos\theta = \frac{4}{5},\ \tan\theta = \frac{3}{4}

similarity & trig on the SAT

Trig ratios stay the same in similar right triangles. You can set up proportions or use tangent, sine, or cosine to solve for missing sides.

For example:

If $\\tan \\theta = \\frac{5}{12}$ in a triangle and you know the adjacent side is 24, then opposite side is $5 \\times (24/12) = 10$ .

Test your right triangle & trig skills!

Question 1

An isosceles right triangle has a perimeter of $94 + 94\sqrt{2}$ inches. What is the length, in inches, of one leg of this triangle?

Question 2

$RS = 440,\ \ ST = 384,\ \ TR = 584$
The side lengths of right triangle $RST$ are given. Triangle $RST$ is similar to triangle $UVW$ , where $S$ corresponds to $V$ and $T$ corresponds to $W$ . What is the value of $\tan W$ ?

Question 3

Triangle $ABC$ is similar to triangle $DEF$ , where $A$ corresponds to $D$ and $C$ corresponds to $F$ . Angles $C$ and $F$ are right angles. If $\tan(A) = \sqrt{3}$ and $DF = 125$ , what is the length of $DE$ ?

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