A closer look at right triangles

The Right Triangle leg² + leg² = hypotenuse² hypotenuse leg leg Acute angles are complementary. Two acute angles The Right Triangle hypotenuse leg One right angle leg

Pythagorean Theorem The Pythagorean Theorem describes the relationship between the sides of a right triangle. leg² + leg² = hypotenuse² A Pythagorean triple is a set of integers, a, b, and c, that could be the sides of a right triangle if a² + b² = c². 3, 4 and 5 are a Pythagorean triple because 3² + 4² = 5² and all three numbers are whole numbers. 7, 8 and 12 are NOT a Pythagorean triple because 7² + 8² = 12² even though they are all whole numbers. 5, 9.5 and √115.25 are NOT a Pythagorean triple - 5² + 9.5² = √115.25 ² BUT the three numbers are not whole numbers.

Many mathematicians over the centuries have developed formulas for generating side lengths for right triangles. Some generate Pythagorean triples, others just generate the side lengths for a right triangle.

Pythagoras' Formula n² - 1 2 n² + 1 2 n , , Use Pythagoras’ formula to find a Pythagorean triple when n is an odd number. Find a Pythagorean triple using Pythagoras’ formula when n = 7.

Plato's Formula a² 4 a² 4 - 1 , a , + 1 Use Plato’s formula to find a Pythagorean triple when a is an even positive integer greater than 2. Find a Pythagorean triple using Plato’s formula when a = 6.

Euclid's Formula x - y 2 x + y 2 , xy , Euclid’s formula won’t always give you a Pythagorean triple. If you restrict values of x and y to either two even or two odd numbers in Euclid’s formula you will at least have two whole numbers. Find the lengths of the sides of a right triangle if x = 7 and y = 9.

Maseres' Formula 2pq , p² - q² , p² + q² p² + q² 2pq p² - q² Maseres wrote many mathematical works which show a complete lack of creative ability. He rejected negative numbers and that part of algebra which is not arithmetic. It is probable that Maseres rejected all mathematics which he could not understand. Use Maseres’ formula to find a Pythagorean triple when you are given two whole numbers. Find a Pythagorean triple using the numbers 9 and 10. p² + q² 2pq p² - q²

Some SPECIAL right triangles

This is a special right triangle called a 45°-45°-90° triangle. Why is it given this name? This is a special right triangle because there is a special relationship between the sides of the triangle. Find the length of the hypotenuse of this triangle. Simplify the radical. 45° 10√2 10 45° 10

Check out these other 45°-45°-90° triangles. Find the length of the missing side – simplify radicals. Can you figure out the special relationship between the sides of a 45°-45°-90° triangle? 45° 1√2 4√2 7√2 1 7 4 1 45° 4 7 45° 12√2 2√2 2 9√2 12 9 45° 2 9 12

What is the special relationship between the lengths of the sides of a 45°-45°-90° triangle? What is the length of each missing side of this triangle? 45° leg√2 leg 45° leg

This is a special right triangle called a 30°-60°-90° triangle. Why is it given this name? This is a special right triangle because there is a special relationship between the sides of the triangle. Find the length of the missing leg of this triangle. Simplify the radical. The legs of this type of triangle are given special names. 60° short leg 20 10 30° long leg 10√3

Check out these other 30°-60°-90° triangles. Find the length of the missing side – simplify radicals. Can you figure out the special relationship between the sides of a 30°-60°-90° triangle? 8 60° 4 60° 2 60° 14 1 30° 7 30° 4√3 1√3 30° 7√3 60° 60° 24 18 12 60° 9 4 2 30° 12√3 30° 30° 9√3 2√3

What is the special relationship between the lengths of the sides of a 30°-60°-90° triangle? What is the length of each missing side of this triangle? 60° 2 (short leg) short leg 30° short leg√3

COLORED NOTE CARD Pythagorean Triples and Special Right Triangles 60° Pythagorean Triple - A set of three whole numbers such that a² + b² = c² Pythagoras’ formula Plato’s formula Euclid’s formula Maseres’ formula n² - 1 2 n² + 1 2 a² 4 a² 4 n , - 1 , a , + 1 , x - y 2 x + y 2 2pq , p² - q² , p² + q² , xy , 45° 60° 2 (short leg) short leg leg√2 leg 30° 45° short leg√3 leg

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