Math

Right Triangles and Trigonometry

2 min readMedium5-question drill

Right triangles show up all over the Math section, and most of their problems come down to just three tools: the Pythagorean theorem, two special triangles, and the SOH-CAH-TOA trig ratios. Memorize those and you can knock out these questions in seconds.

Where this lesson fits

You are here

→Lines, Angles, and Triangles →Circles →Coordinate Geometry

A right triangle is any triangle with one 90° angle (the little square corner). The side opposite that right angle — always the longest side — is the hypotenuse. The other two sides are the legs.

1. The Pythagorean theorem. For any right triangle with legs a and b and hypotenuse c:

a² + b² = c²

The hypotenuse is always c (alone on one side). If you know any two sides, you can solve for the third.

Loading…

A 3-4-5 right triangle: legs a and b, hypotenuse c opposite the right angle.

2. Special right triangles. Two triangles appear so often the test expects you to recognize them instantly:

45-45-90 triangle: the two legs are equal, and the hypotenuse = leg × √2. So the side ratio is 1 : 1 : √2. To go from leg to hypotenuse, multiply by √2; to go backward, divide by √2.
30-60-90 triangle: the sides are in ratio 1 : √3 : 2. The shortest side is opposite the 30° angle, the medium side (× √3) is opposite the 60° angle, and the hypotenuse (× 2, the longest) is opposite the 90° angle.

Loading…

A 30-60-90 triangle with sides in ratio 1 : √3 : 2.

3. Trigonometry — SOH-CAH-TOA. For one of the non-right angles (call it θ), label the sides relative to θ: the opposite side faces it, the adjacent side touches it (and isn't the hypotenuse). Then:

sin θ = opposite / hypotenuse (SOH)
cos θ = adjacent / hypotenuse (CAH)
tan θ = opposite / adjacent (TOA)

One handy fact the test loves: the sine of an angle equals the cosine of its complement. Since the two non-right angles add to 90°, sin(x°) = cos(90° − x°). So if sin A = 3/5, then cos B = 3/5 where A + B = 90°.

Most questions just ask you to plug into one of these three relationships. The hard part is correctly labeling opposite vs. adjacent for the angle in question — so always start by marking your angle and identifying the hypotenuse.

Loading…

Quick check

Check your understanding with a question from this topic:

In a 45-45-90 triangle, the hypotenuse has length 10√2. What is the length of each leg?

Your answer:

Enter a whole number, fraction (e.g. 3/4), or decimal (e.g. .75).

Worked examples

Example 1

A right triangle has legs of length 9 and 12. What is the length of the hypotenuse?

Example 2

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

Example 3

In right triangle ABC, the right angle is at C. If sin A = 7/25, what is the value of cos B?

Common pitfalls

Treating a leg as the hypotenuse

The Pythagorean theorem requires c to be the hypotenuse (opposite the right angle). If a problem gives you the hypotenuse and one leg, you must SUBTRACT: leg² = c² − other leg², not add.

Mixing up the special-triangle ratios

Students confuse √2 (for 45-45-90) with √3 (for 30-60-90). Remember: equal legs → √2; the 30-60-90 has a √3 for its medium side and a 2 for the hypotenuse.

Mislabeling opposite vs. adjacent

Opposite and adjacent are defined relative to the angle you're using. Always circle your angle first; the side touching it (other than the hypotenuse) is adjacent, the side across from it is opposite.

Forgetting the sin/cos complement rule

When a problem gives sin of one angle and asks for cos of another in the same right triangle, check if the angles are complementary — sin x = cos(90 − x) lets you answer instantly without finding any sides.

Key takeaways

Pythagorean theorem: a² + b² = c², where c is always the hypotenuse.
45-45-90 sides are in ratio 1 : 1 : √2; 30-60-90 sides are 1 : √3 : 2.
SOH-CAH-TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj — labeled relative to your chosen angle.
For complementary angles, sin x = cos(90 − x).
Memorize the common Pythagorean triples (3-4-5, 5-12-13, 8-15-17) to skip arithmetic.

Watch & learn

Curated Khan Academy walkthroughs on Right Triangles and Trigonometry. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.

Tracks your progress across lessons.

Try it yourself

5 practice questions on Right Triangles and Trigonometry, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Right Triangles and Trigonometry

You are here

→Lines, Angles, and Triangles →Circles →Coordinate Geometry