Lesson 1: Understanding and Proving the Pythagorean Theorem

Property

In a right triangle, the legs (sides $a$ and $b$) are the two sides that form the right angle. The hypotenuse (side $c$) is the side opposite the right angle and is always the longest side.

Examples

• In a right triangle like $\Delta RST$ (where $\angle S = 90^\circ$), the legs are $\overline{RS}$ and $\overline{ST}$ because they form the 90° angle. The hypotenuse is $\overline{RT}$ because it is the longest side across from the 90° angle.
• A right triangle has side lengths of 8 cm, 15 cm, and 17 cm. The longest side is 17 cm, so it is the hypotenuse. The other two sides, 8 cm and 15 cm, are the legs.
• In a triangle where the sides adjacent to the right angle measure $x$ and $y$, and the third side is $z$, the sides $x$ and $y$ are the legs. The side $z$ is the hypotenuse.

Explanation

Property

For a right triangle whose leg lengths are $a$ and $b$ and whose hypotenuse is of length $c$, the relationship between the side lengths is given by the formula:

Examples

• A right triangle has legs of length $a=5$ and $b=12$. To find the hypotenuse $c$, we use $c^2 = 5^2 + 12^2 = 25 + 144 = 169$. So, $c = \sqrt{169} = 13$.
• A right triangle has a hypotenuse of length $c=10$ and one leg of length $a=6$. To find the other leg $b$, we use $b^2 = c^2 - a^2 = 10^2 - 6^2 = 100 - 36 = 64$. So, $b = \sqrt{64} = 8$.
• The legs of a right triangle are both 3 units long. The hypotenuse $c$ is found with $c^2 = 3^2 + 3^2 = 9 + 9 = 18$. The length of the hypotenuse is $c = \sqrt{18}$.

Explanation

This famous theorem is a secret code for right triangles! It connects the lengths of the two shorter sides (legs) to the longest side (hypotenuse). If you know the lengths of any two sides, you can always find the third.