The Pythagorean Theorem

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The Pythagorean Theorem c a b.

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Presentation transcript:

The Pythagorean Theorem Presented by: Tanya Van Dam Kalles Junior High Algebra 2 Spring 2000

Pythagoras Lived in southern Italy during the sixth century B.C. Considered the first true mathematician Used mathematics as a means to understand the natural world First to teach that the earth was a sphere that revolves around the sun

Right Triangles Longest side is the hypotenuse, side c (opposite the 90o angle) The other two sides are the legs, sides a and b (either leg can be a or b, it actually won’t matter) Pythagoras developed a formula for finding the length of the sides of any right triangle c a b

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

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.

The Pythagorean Theorem, a2 + b2 = c2, relates the sides of RIGHT triangles. a and b are the lengths of the legs c is the length of the hypotenuse.

Find the length of the missing side of the right triangle 1st: label the legs as a and b, label the hypotenuse as c 2nd: plug the values into the Pythagorean theorem a2 + b2 = c2 42 + 32 = c2 3rd: solve the equation for the missing side 16 + 9 = c2 25 = c2 c = 5 cm c a b

Find the length of the missing side of the right triangle 1st: label the legs as a and b, label the hypotenuse as c 2nd: plug the values into the Pythagorean theorem a2 + b2 = c2 92 + b2 = 152 3rd: solve the equation for the missing side 81 + b2 = 225 b2 = 144 b = 12 cm c b a

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?

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

Ladder Problem Solution b = 24 m How did you do? B

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?

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

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

Do NOT Do This!