Pythagorean Theorem & Distance Formula

Anatomy of a right triangle The hypotenuse of a right triangle is the longest side. It is opposite the right angle. The other two sides are legs. They form the right angle. Lesson 7-2: The Pythagorean Theorem2 hypotenuse leg

Lesson 7-2: The Pythagorean Theorem3 leg (a) leg (b) hyp (c) If a triangle is a right triangle, then leg 2 + leg 2 = hyp 2, or a 2 + b 2 = c 2 Pythagorean Theorem **NOTE: It does not matter which leg is labeled a or b. It only matters that the hypotenuse is c.

In the following figure if b = 5 and c = 13, Find a. leg 2 + leg 2 = hyp 2 a = 13 2 a = a 2 = 144 a = 12 Finding the legs of a right triangle: Lesson 7-2: The Pythagorean Theorem4 a b c

5 a b c If, a 2 + b 2 = c 2, then the triangle is a right triangle. C must be the LONGEST side! Pythagorean Theorem CONVERSE

Given three lengths of 6, 8, 10, describe the triangle. Plug in lengths to Pythagorean Theorem: a 2 + b 2 and c 2 Since 100 = 100, this is a right triangle. a 2 + b 2 c = Given the lengths of three sides, how do you know if you have a right triangle? Lesson 7-2: The Pythagorean Theorem6 a b c = = =

The Converse of the Pythagorean Theorem Lesson 7-2: The Pythagorean Theorem7 a b c If a 2 + b 2 > c 2, then the triangle is acute. The longest side is too short!

Lesson 7-2: The Pythagorean Theorem8 ab c If a 2 + b 2 < c 2, then the triangle is obtuse. The longest side is too long! The Converse of the Pythagorean Theorem

Instead of drawing a right triangle and using the Pythagorean Theorem, we can use the following formula: distance = where (x 1, y 1 ) and (x 2, y 2 ) are the ordered pairs corresponding to the two points. So let’s go back to the example. Distance Formula

Example x y Find the distance between these two points. Solution: First : Find the coordinates of each point. ? – 2 – 4 (– 4, – 2) 8 3 (8, 3)

Example x y Find the distance between these two points. Solution: First: Find the coordinates of each point. (x 1, y 1 ) = (-4, -2) (x 2, y 2 ) = (8, 3) ? (– 4, – 2) (8, 3)

Solution cont. Then: Since the ordered pairs are (x 1, y 1 ) = (-4, -2) and (x 2, y 2 ) = (8, 3) Plug in x 1 = -4, y 1 = -2, x 2 = 8 and y 2 = 3 into distance= = = = 13 Example cont.