Pythagorean Theorem

Pythagoras (puh thag or us) was a Greek philosopher and mathematician, born in Samos in the sixth century, B.C. He and his followers tried to explain everything with numbers. We remember him today mainly for his equation relating the lengths of the legs of a right triangle to the length of its hypotenuse.

The Pythagorean (puh thag or ee un) Theorem, also called the Pythagorean Property, says that the sum of the squares of the lengths of the legs of any right triangle is equal to the square of the length of the hypotenuse, a² + b² = c².

Another way of looking at the Pythagorean Theorem is to think about actual squares of the lengths of the sides of a right triangle.

Pythagorean Triples are groups of three whole numbers that make the Pythagorean Theorem true (and therefore define a true right triangle). 3, 4, and 5 are a Pythagorean Triple. There are several ways to generate Pythagorean Triples. Here are two:

• Use multiples of known triples (3, 4, 5; 6, 8, 10; 9, 12, 15; etc.)
• Choose two whole numbers, e and f. Be sure that f is greater than e. Use these formulas to find triples:

You can appropriately ask students to use the Pythagorean Theorem

1. to check whether a triangle is a right triangle;
2. to find the missing length of leg or hypotenuse of a given right triangle;
3. to find the length of a diagonal of a square or rectangle;
4. to find the length of a side of a square given the diagonal;
5. to find the length of a side of a rectangle given the diagonal and the other side.

When you work with the Pythagorean Theorem, numbers may get quite messy. Have your students compute without units, then attach the units after they find the numerical part of the answer.

There are two ways to show answers to a problem involving square roots. For example, find the length of the hypotenuse for this right triangle.

You have values for a and b in the equation a² + b² = c².
Substitute these values into the equation and solve for c. (2²) + (4²) = c²
4 + 16 = c²
20 = c²
20 = c
= c
25 = c
4.47214 c

Both the simplified radical form and the approximation for the answer to the computation should be acceptable answers. In radical form, the answer is exact. However, if the value under the radical is not a perfect square, its square root is approximate and must carry the wavy equals symbol. Most calculators will give approximations for square roots. If you do not have calculators available, use a table of square roots.

The answer to the original problem is: The hypotenuse is 25 centimeters long or The hypotenuse is about 4.5 centimeters long.

How to Simplify Inside a Radical

If a number in radical form is not a perfect square, you may be able to simplify that number by factoring. Here are some ways you can show answers to problems involving square roots.

= 169 = 13 because 169 is a perfect square.

= 125

Since 125 is not a perfect square, its square root is not a rational number. However, 125 = 25 5 and 25 IS a perfect square, so you can say 125 = . While it's NOT OK to write sums or differences under one radical as a sum or difference of two roots, it IS OK to show products or quotients under one radical as a product or quotient of two roots. This means that 125 = = 25 5, or 55.

= 41

Since 41 is not a perfect square and does not have any factors that are perfect squares, 41 cannot be simplified.