PYTHAGOREAN THEOREM

Concept: A right triangle has two sides that meet in a right angle. The side opposite the right angle is called the hypotenuse and is the longest side of a right triangle. In a right triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. We call this statement the Pythagorean Theorem. In symbols this can be expressed a2 + b2 = c2. Any time you know the measurements for two sides of a right triangle, you can solve for the third by using the Pythagorean Theorem.

Steps for Using the Pythagorean Theorem

Step 1 | Write the formula for the Pythagorean Theorem.
Step 2 | Substitute the known values.
Step 3 | Solve.
Step 4 | Write the final answer.

Example 1 Solve for side a

Step 1 | Write the formula for the Pythagorean Theorem. | a2 + b2 = c2
Step 2 | Substitute the known values. | a2 + 82 = 8. 2462
Step 3 | Solve. In solving this equation we must use some basic algebra. We must isolate the variable on one side of the equal sign, and the numbers on the other side. To accomplish this in this equation, begin by solving for 82, which is 64. Next, isolate the variable (a2) by subtracting 64 from the left side of the equation. 64 subtracted from 64 is 0, which will leave only a2 on the left side of the equation. We must perform the same operation on both sides of the equation, so we must subtract 64 from 68, which will leave 4. This results in an equation which reads a2 = 4. To find a, we must find the square root of 4, which is 2. Our final answer is a = 2. | a2 + 82 = 8. 2462 a2 + 64 = 68 a2 + 64 = 68 - 64 - 64 a2 + 0 = 4 a2 = 4 a = a = 2
Step 4 | Write the final answer. | a = 2

Example 2 Solve for side C.

Step 1 | Write the formula for the Pythagorean Theorem. | a2 + b2 = c2
Step 2 | Substitute the known values. | 32 + 52 = c2
| Visualize the above triangle as two identical right triangles placed back to back, as shown in the box at the right. The dotted black line in the triangle above is side b for both triangles at the right. Side b measures 5. |
| The base of the original triangle measures 6. Side a in both right triangles is half of the original base. Side a in both right triangles will measure 3, half of 6. Thus, in this equation, a = 3 and b = 5 |
Step 3 | Solve. | a2 + b2 = c2 32 + 52 = c2 9 + 25 = c2 34 = c2 = c 5.83 = c
Step 4 | Write the final answer. | c = 5.83