Before you read

Answer first — see what you already know.

A right-triangular support beam has legs of 9 ft and 12 ft. What is the length of the hypotenuse?

A
13
B
15
C
18
D
21

Check your answer

Answer:

Before you read

Answer first — see what you already know.

A 10-foot ladder leans against a warehouse wall, with its base 6 feet from the wall. How high up the wall does the ladder reach?

A
4 ft
B
6 ft
C
8 ft
D
$136$ ft

Check your answer

Answer:

Theory

The Pythagorean Theorem

In a right triangle with legs $a$ and $b$ and hypotenuse $c$ :
$a^2 + b^2 = c^2$

The hypotenuse is always the longest side and is opposite the right angle.

Common Pythagorean triples (memorize these!):
- $3, 4, 5$ (and multiples: $6, 8, 10$ ; $9, 12, 15$ ; etc.)
- $5, 12, 13$
- $8, 15, 17$
- $7, 24, 25$

The 3-4-5 Pythagorean triple

The 5-12-13 Pythagorean triple

The 8-15-17 Pythagorean triple

Converse: If $a^2 + b^2 = c^2$ , the triangle is a right triangle.

Example 1

A right triangle has legs 6 and 8. Find the hypotenuse.

$c^2 = 6^2 + 8^2 = 36 + 64 = 100$

$c = 10$

c = 10

Strategy

SAT Pro Tip

If you see a right triangle on the SAT, always check if it's a multiple of a Pythagorean triple. $5, 12, 13$ and $3, 4, 5$ appear constantly. A triangle with legs 15 and 20 has hypotenuse 25 (it's $3, 4, 5$ $\times$ 5). Recognizing triples is faster than computing.

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