Pythagorean Theorem Cheat Sheet

The Formula

a² + b² = c², where 'a' and 'b' are the lengths of the legs of a right triangle, and 'c' is the length of the hypotenuse.

Example: If a=3 and b=4, then 3² + 4² = c², so 9 + 16 = c², 25 = c², and c = 5.

Right Triangle

The Pythagorean Theorem only applies to right triangles, which have one 90-degree angle.

Example: A triangle with angles 30°, 60°, and 90° is a right triangle.

Hypotenuse

The hypotenuse is the side opposite the right angle. It is always the longest side of the right triangle.

Example: In a right triangle with legs of length 3 and 4, the hypotenuse has a length of 5.

Legs

The legs are the two sides that form the right angle.

Example: In a right triangle with a hypotenuse of length 5 and one leg of length 3, the other leg has a length of 4.

Finding the Hypotenuse (c)

If you know the lengths of the legs (a and b), use the formula c = √(a² + b²).

Example: If a=6 and b=8, then c = √(6² + 8²) = √(36 + 64) = √100 = 10.

Finding a Leg (a or b)

If you know the length of the hypotenuse (c) and one leg (e.g., a), use the formula b = √(c² - a²).

Example: If c=13 and a=5, then b = √(13² - 5²) = √(169 - 25) = √144 = 12.

Rearranging the Formula

The formula can be rearranged to solve for any side: a² = c² - b² or b² = c² - a².

Example: To find 'a' when b=8 and c=10: a² = 10² - 8² = 100 - 64 = 36, so a = √36 = 6.

Distance Between Two Points

The Pythagorean Theorem can be used to find the distance between two points on a coordinate plane. Treat the horizontal and vertical distances as legs of a right triangle.

Example: Points (1, 2) and (4, 6). a = 4-1 = 3, b = 6-2 = 4. c = √(3² + 4²) = 5.

Real-World Problems

Many real-world problems involving right triangles can be solved using the Pythagorean Theorem, such as finding the height of a ladder against a wall.

Example: A 10-foot ladder leans against a wall, with its base 6 feet from the wall. The height the ladder reaches is √(10² - 6²) = 8 feet.

Isosceles Right Triangles

In an isosceles right triangle (45-45-90 triangle), the legs are equal in length. If the leg length is 'a', the hypotenuse is a√2.

Example: If a leg of an isosceles right triangle is 5, the hypotenuse is 5√2.