Pythagorean Theorem

a2 + b2 = c2 The variable a and b represent the legs of a right triangle. Legs – the two sides that form the right angle.

We call it a right triangle because it contains a right angle.

The side across from the right angle is called the hypotenuse. a c a b

The variable c represents the hypotenuse. The hypotenuse is the longest side of a right triangle, across from the right angle.

a2 + b2 = c2 In any right triangle, the sum of the squares of the lengths of the 2 legs is equal to the square of the length of the hypotenuse.

Finding the length of c Label the sides Substitute the lengths for the appropriate variables in the Pythagorean theorem. Simplify Solve for c *The inverse of a square (x2) is the square root (√x). C = √8 2 2

Find the length of a diagonal of the rectangle: 15" 8" ?

Find the length of a diagonal of the rectangle: 15" 8" ? b = 8 a = 15 c

Find the length of a diagonal of the rectangle: 15" 8" ? b = 8 a = 15 c

Finding a or b 12 Label the sides Substitute the lengths for the appropriate variables Simplify Solve for variable Subtract from both sides Find the square root 15 9 12

Practice using the Pythagorean Theorem to solve these right triangles: 10 b 26

10 b 26 (a) (c)

Ladders 15 b 5 Draw a picture: A 15 foot ladder leans up against a building. The foot of the ladder is 5 feet from the base of the building. How high up the wall, to the nearest foot does the ladder reach? Draw a picture: 15 b 5

Solving the problem a2 + b2 = c2 Write formula x2 + 52 = 152 Substitute in x2 + 25 = 225 Solve for x - 25 -25 x2 = 200 x = 14.142135 Check: How am I to leave my answer? The ladder reaches 14 feet up the wall.

Identifying a right triangle The converse of the Pythagorean theorem says that if a triangle has side lengths a, b, and c and a2 + b2 = c2, then the triangle is a right triangle. 7,24,25 Yes 625 = 625 5,8,12 No 89 ≠ 144

Solving the problem A soccer field is a rectangle 90 meters wide and 120 meters long. The coach asks players to run from one corner to the corner diagonally across the field. How far do the players run? 120 m 90m c = 150m

Solving the problem How far from the base of the house do you need to place a 15’ ladder so that it exactly reaches the top of a 12’ wall? 15’ 12’ b

Solving the problem What is the length of the diagonal of a 10 cm by 15 cm rectangle?

Solving the problem An isosceles triangle has congruent sides of 20 cm. The base is 10 cm. What is the height of the triangle?

Solving the problem A baseball diamond is a square that is 90’ on each side. If a player throws the ball from 2nd base to home, how far will the ball travel? 90’ 90’ c

Solving the problem Jill’s front door is 42” wide and 84” tall. She purchased a circular table that is 96 inches in diameter. Will the table fit through the front door?

Solving the problem Daniel rides his bicycle 21 km west and then 18 km north. How far is he from his starting point?

Solving the problem Find the width of a triangle that has a 15cm height and a hypotenuse of 19 cm.

Solving the problem John is trying to determine the length of the staircase he will need for a deck that is 12 feet high. He wants to start the stairs 21 feet from the deck. How long does the staircase need to be?

Solving the problem The diagonal of a rectangle is 25 in. The width is 15 inches. What is the length?

Solving the problem You're locked out of your house and the only open window is on the second floor, 25 feet above the ground. You need to borrow a ladder from one of your neighbors. There's a bush along the edge of the house, so you'll have to place the ladder 10 feet from the house. What length of ladder do you need to reach the window?

Solving the problem When Joe drives to school, he picks up his friend Dave. Dave lives directly south of Joe. Their school is directly east of Joe's house. Joe usually drives an average of 30 miles per hour. If Joe lives 12 miles from Dave, and Dave lives 20 miles from school, how much time would Joe save if he did not have to pick up his friend Dave?

Solving the problem You have a wooden box that measures 4 ft. by 3 ft. by 2 ft. What is the measure of x?

Solving the problem