The right triangle has been used in trades for thousands of years. Ancient
Egyptians found that they could always get a square corner using the 3-4-5 right
triangle. (Carpenters still use the 3-4-5 triangle to square corners.) Later, a Greek
mathematician named Pythagoras developed a formula to find the side lengths for any right triangles.

Pythagoras treated each side of

a right triangle as if it was a

side of a square. He found that

the total area of the two smaller

squares was equal to the area of

the largest square for every right

triangle.
The area of any square is found

by squaring one of the sides.

The areas of these squares are

are 3^2 , 4^2 , and 5^2 . If the

two shorter sides are squared

and added together, the answer

equals the longer side squared:

The three sides of a **right triangle** are represented by the variables **a, b,** and **c.** The variable **c** *always * represents the longest side. You can write the formula used
above as .

This is the formula of the Pythagorean Theorem.

The longest side of a right triangle is *always * directly across from the 90 degree angle. This side is called the **hypotenuse.**

The other two sides are often called

the **legs.** For now, it really doesn't

matter which one of the legs you call

**a** or **b,** as long as you make one **a**

and the other **b.**

Because the 3-4-5 right triangle is simple to work with,

let's use it to show how the Pythagorean formula

verifies that a triangle is a right triangle.
Since the numbers on both sides of the = mark are the same, the 3-4-5 triangle is a right triangle.

**Please note: The Pythagorean formula is a rule!** To verify that a triangle is a right triangle, use the formula.

If the numbers on each side of the = sign are equal, then the triangle is a right triangle.

**Right triangle practice:** Check these triangles to see which ones are right triangles.

a b c a b c
(1) 6 8 10 (6) 9.99 13.32 16.65
(2) .48 .64 .8 (7) 4.5 6 7.5
(3) 1 1 1 (8) 7 10 13
(4) 8 6 10 (9) 9 2.5 13
(5) 120 160 200 (10) 2 2.6666 3.3333

**Answers.**

**Note:**

**
** Lower case letters (a, b, c)

are used as variables for the sides.

Upper case letters (A, B, C)

are used for the angles opposite

those sides.