Pythagorean Theorem, its Converse and the coordinate system Objective: Students will use the Pythagorean theorem and its converse.

Algebra Standards: 2.0 Students understand and use such operations as taking the opposite, reciprocal, raising to a power, and taking a root. This includes the understanding and use of the rules of exponents.

Vocabulary: For Right Triangles Only! c b a Pythagorean Theorem c b a hypotenuse - always opposite the right angle c leg b a Pythagorean Theorem leg a2 + b2 = c2 hypotenuse c b a a b

Pythagorean Theorem Pythagorean Theorem 2 a) b) 5 5 12 a2 + b2 = c2 a2 #1 Use the Pythagorean the square Find the missing side. 2 a) b) 5 5 Pythagorean Theorem 12 Pythagorean Theorem a2 + b2 = c2 a2 + b2 = c2 22 + 52 = x2 52 + 122 = x2 4 + 25 = x2 25 + 144 = x2 29 = x2 169 = x2 29 = |x| 13 = |x| x = 29 or -29 x = 13 or -13

Pythagorean Theorem Pythagorean Theorem 3 c) d) 5 3 7 a2 + b2 = c2 a2 #1 Use the Pythagorean the square Find the missing side. 3 c) d) 5 3 7 Pythagorean Theorem Pythagorean Theorem a2 + b2 = c2 a2 + b2 = c2 32 + x2 = 72 32 + x2 = 52 9 + x2 = 49 9 + x2 = 25 – 9 –9 – 9 –9 x2 = 40 x2 = 16 |x| = 40 = 2 10 |x| = 4 x = 2 10 or -2 10 x = 4 or -4

Pythagorean Theorem 10 a2 + b2 = c2 x x2 + (x+2)2 = 102 x x2 #2 Use the Pythagorean the square A right triangle has one leg that is 2 inches longer that the other leg. The hypotenuse is 10 inches. Find the unknown legs. Pythagorean Theorem 10 a2 + b2 = c2 x x2 + (x+2)2 = 102 x x2 + x2 + 4x + 4 = 100 + 2 2x2 + 4x – 96 = 2 x = 6 or – 8 x2 + 2x – 48 = Leg one = 6 Leg two = 8 ( )( ) x – 6 x + 8 =

Pythagorean Theorem 15 a2 + b2 = c2 x x2 + (x+3)2 = 152 x x2 #3 Use the Pythagorean the square A right triangle has one leg that is 3 inches longer that the other leg. The hypotenuse is 15 inches. Find the unknown legs. Pythagorean Theorem 15 a2 + b2 = c2 x x2 + (x+3)2 = 152 x x2 + x2 + 6x + 9 = 225 + 3 2x2 + 6x – 216 = 2 x = 9 or – 12 x2 + 3x – 108 = Leg one = 9 Leg two = 12 ( )( ) x – 9 x + 12 =

Pythagorean Theorem 2 a2 + b2 = c2 22 + 22 = x2 2 4 + 4 = x2 8 = x2 2 #4 Use the Pythagorean the square A board game is a square 2 ft by 2ft. What is the length of the diagonal? Pythagorean Theorem 2 a2 + b2 = c2 22 + 22 = x2 2 4 + 4 = x2 8 = x2 2 2 2 = |x| 2 x = 2 2 or -2 2

Determine whether the given lengths could #5 Converse of the Pythagorean Theorem Determine whether the given lengths could be the sides of a right triangle. 1) 5 12 13 2) 9 12 6 a2 + b2 = c2 a2 + b2 = c2 52 + 122 = 132 92 + 62 = 122 25 + 144 = 169 81 + 36 = 144 169 = 169 117 = 144 yes no

A car drives 20 miles due east and then 45 miles Real world Application A car drives 20 miles due east and then 45 miles due south. To the nearest hundredth of a mile, how far is the car from its starting point? a2 + b2 = c2 20 miles 202 + 452 = x2 400 + 2025 = x2 x 45 miles 2425 = x2 = |x| x = or

Real world Application

Real world Application

Pythagorean Theorem and the Coordinate Plane Standard: 8.G8 – Apply the Pythagorean Theorem to find the distance Between two points in a coordinate system

Using Coordinates  

Find the distance between the two points. Point A (-5, -6) and Point B (-2, 4)