Pythagorean Pythagorean TheoremTheorem

Right TriangleHypotenuse

LegsPythagorean Theorem

Radical/RadicandSquare Root

VOCABULARY

RIGHT TRIANGLE

Longest side is the hypotenuse, side c (opposite the 90o angle)

The other two sides are the legs, sides a and b

Pythagoras developed a formula for finding the length of the sides of any right triangle

RADICAL/RADICAND

Definition: A number the produces a specified quantity when multiplied by itself.

SQUARE ROOT

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.”

aa22 + b + b22 = c = c22

In different words… If the angle opposite the hypotenuse is a

right angle, then aa22 + b + b22 = c = c22

If two measures of the sides of a right triangle

are known, the Pythagorean Theorem can be used to find the measure of the third

side.

EXAMPLE #1

152+82=c2

225+64=c2

289=c2

17=c

EXAMPLE #2

YOUR TURN! ☺

Normal: If x, then y. Converse: If y, then x.

Example:If it is raining, then the grass is wet.Converse: If the grass is wet, then it is raining.

CONVERSE?

“If a triangle has sides of lengths a, b, and c; and aa22 + b + b22 = c = c22

then the triangle is a right triangle with

hypotenuse of length c.”

CONVERSE OF THE PYTHAGOREAN THEOREM

You can determine if a triangle is a right triangle if

the 3 side lengths fit into aa22 + b + b22 = c = c22

EXAMPLE #1

52+72=92

25+49=81?

No! This is not a right triangle.

YOUR TURN! ☺

COMMON PYTHAGOREAN TRIPLES

Find the length of the hypotenuse if1. a = 12 and b = 16.

122 + 162 = c2

144 + 256 = c2

400 = c2

Take the square root of both sides.

2400 c20 = c

Find the length of the hypotenuse if2. a = 5 and b = 7.

Find the length of the hypotenuse given a = 6 and b = 12

1. 1802. 3243. 13.424. 18

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, if4. c = 10 and b = 7.

Find the length of the missing side given a = 4 and c = 5

1. 12. 33. 6.44. 9

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.

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?

REAL-LIFE APPLICATIONS

Baseball Problem

A baseball “diamond” is really a square.

You can use the Pythagorean theorem to find distances around a baseball diamond.

The distance between consecutive bases is 90feet. How far does a catcher have to throwthe ball from home plate to second base?

25

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: aa22 + b + b22 = c = c22

26

BASEBALL PROBLEM

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

27

BASEBALL PROBLEMSOLUTION

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?

28

LADDER PROBLEM

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.

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LADDER PROBLEMSOLUTION

Distance from house: 7 meters

72 + b2 = 252

49 + b2 = 625b2 = 576b = 24 m

How did you do?

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LADDER PROBLEMSOLUTION

B

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DO NOT DO THIS!