Pythagorean Theorem

What is the Pythagorean Theorem?

the first official proof of the Pythagorean Theorem. Pythagoras Pythagoras was a Greek philosopher known as the first “true” mathematician. He is credited with the first official proof of the Pythagorean Theorem.

THE PYTHAGOREAN THEOREM “For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.” a2 + b2 = c2 In different words… If the angle opposite the hypotenuse is a right angle, then a2 + b2 = c2

a AND b REPRESENT THE LEGS. THE THEOREM a AND b REPRESENT THE LEGS. c REPRESENTS THE HYPOTENUSE (THE LONGEST SIDE).

RIGHT TRIANGLE VOCABULARY Longest side is the hypotenuse, side c (opposite the 90o angle) The other two sides are the legs, sides a and b THE THEOREM ONLY WORKS FOR RIGHT TRIANGLES.

Example #1 A right isosceles triangle has legs 6 meters long each. Find the length of the hypotenuse to the nearest tenth of a meter. a2 + b2 = c2 ____2 + ____2 = ____2

Example #2 a2 + b2 = c2 ____2 + ____2 = ____2

YOUR TURN! ☺

Find the length of the diagonal.

Normal: If x, then y. Converse: If y, then x. CONVERSE? Example: If it is raining, then the grass is wet. Converse: If the grass is wet, then it is raining.

CONVERSE OF THE PYTHAGOREAN THEOREM The theorem works both ways! “If a triangle is a right triangle, then a2 + b2 = c2. If a2 + b2 = c2, then the triangle is a right triangle. You can determine if a triangle is a right triangle if the 3 side lengths fit into a2 + b2 = c2

Is this a right triangle? Side lengths: 8 cm, 10 cm, 16 cm No! This is not a right triangle. 82+102=162 64+100=256?

YOUR TURN! ☺

On a coordinate plane??? Find the distance between the points (4, -3) and (-2, 1) on the coordinate plane.

COMMON PYTHAGOREAN TRIPLES

Find the length of the hypotenuse if 1. a = 12 and b = 16. 122 + 162 = c2 144 + 256 = c2 400 = c2 Take the square root of both sides. 20 = c

Find the length of the hypotenuse if 2. a = 5 and b = 7. 18

Find the length of the hypotenuse given a = 6 and b = 12 180 324 13.42 18

Find the length of the leg, to the. nearest hundredth, if 3 Find the length of the leg, to the nearest hundredth, if 3. a = 4 and c = 10.

Find the length of the leg, to the nearest hundredth, if 4 Find the length of the leg, to the nearest hundredth, if 4. c = 10 and b = 7.

Find the length of the missing side given a = 4 and c = 5 1 3 6.4 9

5. The measures of three sides of a triangle are given below 5. The measures of three sides of a triangle are given below. Determine whether each triangle is a right triangle. 5 , 3, and 8

The rectangular bottom of a box has an interior length of 19 inches and an interior width of 14 inches. A stick is placed in the box along the diagonal of the bottom of the box. Which measurement is closest to the longest possible length for the stick? a2 + b2 = c2 F 12.8 in. G 16.3 in. H 23.6 in. J 33.0 in.

REAL-LIFE APPLICATIONS The Pythagorean theorem has far-reaching ramifications in other fields (such as the arts), as well as practical applications. The theorem is invaluable when computing distances between two points, such as in navigation and land surveying. Another important application is in the design of ramps. Ramp designs for handicap-accessible sites and for skateboard parks are very much in demand. Can you think of other ways the Pythagorean Theorem can be extremely valuable?

Baseball Problem A baseball “diamond” is really a square. You can use the Pythagorean theorem to find distances around a baseball diamond.

BASEBALL PROBLEM The distance between consecutive bases is 90 feet. How far does a catcher have to throw the ball from home plate to second base? 27

BASEBALL PROBLEM To use the Pythagorean theorem to solve for x, find the right angle. Which side is the hypotenuse? Which sides are the legs? Now use: a2 + b2 = c2 28

BASEBALL PROBLEM SOLUTION The hypotenuse is the distance from home to second, or side x in the picture. The legs are from home to first and from first to second. Solution: x2 = 902 + 902 = 16,200 x = 127.28 ft 29

LADDER PROBLEM A ladder leans against a second-story window of a house. If the ladder is 25 meters long, and the base of the ladder is 7 meters from the house, how high is the window? 30

LADDER PROBLEM SOLUTION First draw a diagram that shows the sides of the right triangle. Label the sides: Ladder is 25 m Distance from house is 7 m Use a2 + b2 = c2 to solve for the missing side. Distance from house: 7 meters 31

LADDER PROBLEM SOLUTION b = 24 m How did you do? B 32

EXAMPLE 1

EXAMPLE 2

EXAMPLE 3 Laramie leaves the trailhead and hikes 5 miles east and 9 miles north to a waterfall. What is the closest distance from the trailhead to the waterfall? Draw a picture to help solve.

EXAMPLE 4 Hailey uses a straight piece of wood that is 8 feet long to prop up an old fence. If the fence is 6 feet tall, how far from the fence is the bottom of the piece of wood? Draw a picture to help solve.

EXAMPLE 5 A right triangle in the coordinate plane has vertices at (0, 6), (8, 0), and (0, 0). What is the length of the hypotenuse?

EXAMPLE 6 To the nearest tenth, what is the distance from R to T?

EXAMPLE 7

EXAMPLE 8 The triangle on the grid represents a section of Miesha’s backyard. What is the length of the hypotenuse?

EXAMPLE 9 41

EXAMPLE 10 42

EXAMPLE 11 43