Right Triangles
Definition, Criteria, Pythagorean Theorem & Special Triangles
SAT Math — Geometry
SAT Tip: Right triangle questions appear on nearly every SAT. Master this lesson and you'll pick up easy points fast.
1. Introduction — What Is a Right Triangle?
Definition
A right triangle is a triangle that contains exactly one angle equal to 90° (a right angle).
Parts of a Right Triangle
| Part | Description |
|---|---|
| Legs ($a$ and $b$) | The two sides that form the right angle |
| Hypotenuse ($c$) | The side opposite the right angle — always the longest side |
| Right-angle symbol | A small square drawn in the corner indicates the 90° angle |
Key Facts
- The two acute angles are complementary — they sum to 90°.
- The hypotenuse is always opposite the largest angle (90°), so it is always the longest side.
Why right triangles matter on the SAT:
• Foundation of trigonometry (SOH CAH TOA)
• Appear in distance, area, diagonal, ladder, shadow problems
• Two special right triangles (45-45-90 and 30-60-90) are given on the SAT reference sheet — but memorizing them saves time!
• Foundation of trigonometry (SOH CAH TOA)
• Appear in distance, area, diagonal, ladder, shadow problems
• Two special right triangles (45-45-90 and 30-60-90) are given on the SAT reference sheet — but memorizing them saves time!
Interactive visualization — available in the full lesson
Interactive diagram — the small green square marks the 90° angle.
2. The Pythagorean Theorem
$$a^2 + b^2 = c^2$$
where $a$ and $b$ are legs and $c$ is the hypotenuse.
Finding the Hypotenuse
$$c = \sqrt{a^2 + b^2}$$
Finding a Leg
$$a = \sqrt{c^2 - b^2} \qquad b = \sqrt{c^2 - a^2}$$
⚠️ Common Mistake: The hypotenuse $c$ must be alone on one side. Never put the hypotenuse under the addition — it's $a^2+b^2=c^2$, not $a^2+c^2=b^2$.
Worked Examples
| Given | Find | Solution |
|---|---|---|
| Legs 3, 4 | Hypotenuse | $c=\sqrt{9+16}=\sqrt{25}=5$ |
| Legs 5, 12 | Hypotenuse | $c=\sqrt{25+144}=\sqrt{169}=13$ |
| Legs 8, 15 | Hypotenuse | $c=\sqrt{64+225}=\sqrt{289}=17$ |
| Leg 6, hyp 10 | Other leg | $a=\sqrt{100-36}=\sqrt{64}=8$ |
| Leg 9, hyp 15 | Other leg | $a=\sqrt{225-81}=\sqrt{144}=12$ |
Interactive visualization — available in the full lesson
5-12-13 right triangle
Converse of the Pythagorean Theorem
If the sides of a triangle satisfy $a^2+b^2=c^2$ (where $c$ is the longest side), then the triangle is a right triangle.
| Test | Result | Triangle Type |
|---|---|---|
| $a^2+b^2 = c^2$ | Equal | Right |
| $a^2+b^2 > c^2$ | Sum is larger | Acute |
| $a^2+b^2 < c^2$ | Sum is smaller | Obtuse |
Example: Sides 6, 8, 10 → $36+64=100=10^2$ ✓ → Right triangle
Example: Sides 5, 6, 8 → $25+36=61 \neq 64=8^2$ → Not a right triangle (and $61<64$ → obtuse)
Example: Sides 7, 8, 10 → $49+64=113>100$ → Acute triangle