Right triangles are the gateway to half the SAT geometry section — Pythagorean theorem, special triangles, and trigonometry all build on the same setup: a triangle with a 90° angle.
The four most common Pythagorean triples — and their multiples
Base triple
×2
×3
Watch for
3 – 4 – 5
6 – 8 – 10
9 – 12 – 15
Most common; small numbers
5 – 12 – 13
10 – 24 – 26
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Mid-size; legs differ widely
8 – 15 – 17
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Larger; appears in coord geom
7 – 24 – 25
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—
Rare but tested
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The two special right triangles — memorize these ratios
Triangle
Side ratio
If short side = $s$…
45-45-90 (isosceles right)
$1 : 1 : \sqrt{2}$
Legs both $s$; hypotenuse $s\sqrt{2}$
30-60-90
$1 : \sqrt{3} : 2$
Short leg $s$; long leg $s\sqrt{3}$; hypotenuse $2s$
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Quick check
Quick check. Identify the hypotenuse (opposite the right angle), use Pythagoras or — if you spot a triple — write the third side directly.
A right triangle has legs of length 6 and 8. What is the hypotenuse?
Worked examples
Example 1
A 13-foot ladder leans against a wall, with its base 5 feet from the wall. How high up the wall does the ladder reach?
Example 2
An equilateral triangle has side length 6. What is its height?
Common pitfalls
Forgetting which side is the hypotenuse
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Mixing up special triangle ratios
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Not recognizing Pythagorean triples
If you see a problem with 5 and 13, the third side is almost certainly 12. With 8 and 17, expect 15. Memorize 3-4-5, 5-12-13, 8-15-17, 7-24-25 and their multiples.
Squaring negatives wrong
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Key takeaways
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Memorize the triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25 — and their multiples.
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Tracks your progress across lessons.
Try it yourself
5 practice questions on Right Triangles, drawn from the question bank. The tutor is one click away if you get stuck.
A right triangle has one 90° angle. The side opposite the right angle (the longest side) is the hypotenuse; the other two sides are the legs.
The Pythagorean theorem is the foundation: for legs a, b and hypotenuse c,
a2+b2=c2
Given any two sides, you can find the third.
Memorize these Pythagorean triples — they show up constantly on the SAT:
3–4–5 (and multiples like 6–8–10, 9–12–15)
5–12–13 (and 10–24–26)
8–15–17
7–24–25
If you see two of those numbers in a problem, you don't even need to compute — recognize the triple and write down the third.
Two special right triangles with non-integer sides — also memorize:
45°–45°–90° (isosceles right triangle): legs are equal, hypotenuse = leg × 2.
legs:leg:hypotenuse=1:1:2
30°–60°–90°: short leg : long leg : hypotenuse = 1:3:2. The short leg is opposite the 30°.
short:long:hypotenuse=1:3:2
SAT-typical setups:
A ladder leans against a wall — Pythagorean triple.
A square's diagonal — 45-45-90 triangle.
A cube or rectangular box — Pythagorean theorem twice (face diagonal then space diagonal).
A point on a coordinate grid — distance is hypotenuse of a right triangle whose legs are the coordinate differences.
Distance formula is just Pythagoras in coordinates:
d=(x2−x1)2+(y2−y1)2
Quick test: if a triangle problem gives you 5 and 12, expect 13. If it gives 8 and 15, expect 17. Recognizing triples saves seconds.
The hypotenuse is the side OPPOSITE the right angle — the longest side. In Pythagoras, it's always c in a2+b2=c2. Mixing up sides flips the equation.
45-45-90: 1:1:2. 30-60-90: 1:3:2 (short : long : hyp). Mixing these costs questions. The mnemonic: 30-60-90 is not symmetric, so the legs differ; 45-45-90 is symmetric, so the legs match.
(x2−x1)2 — if the difference is −3, (−3)2=9, NOT −9. The distance formula always gives a positive distance because squares are positive. Don't drop signs and produce a wrong answer.
Pythagorean theorem: a2+b2=c2, where c is the hypotenuse (opposite the right angle).
45-45-90 sides: 1:1:2 (legs equal, hypotenuse leg × √2).