513 Pythagorean Worked Examples

The Pythagorean Theorem

Name ___________________________ Period ____

One of the most famous sailboat races in the world is the America’s Cup. In each sailboat, the mast and boom form a right angle. The mainsail is in the shape of a right triangle.

In a right triangle, the side opposite the right angle is called the hypotenuse. This side is always the longest side of a right triangle. The other sides are called legs of the triangle.

To find the length of any side of a right triangle when the other two lengths are known, you can use the formula designed by the Greek mathematician Pythagoras.

mainsail

mast

boom hypotenuse

leg

leg

The Pythagorean Theorem: If a and b are the measures of the legs of a right triangle and c is the measure of the hypotenuse, then a2 + b2 = c2.

Find the length of the hypotenuse of a right triangle if a=12 in. and b=5 in.

a2 + b2 = c2 Pythagorean Theorem

122 + 52 = c2 Substitute a=12 in., b=5 in.

144 + 25 = c2 Simplify

169 = c2 Simplify by addition

±

!

169 =

!

c 2 Take the positive square root of each side

13 in. = c Disregard the negative square root ( -13). Why? The hypotenuse is a distance, and therefore cannot be a negative number.

Find the length of the hypotenuse of a right triangle if a=8 m and b=6 m.


82 + 62 = c2 Substitute a=8m, b=6m



!

100 =

!

c 2 Take the square root of each side

_____ m = c How long is the hypotenuse?

EXAMPLE 1 – Finding the Hypotenuse

a

b c


c

b

a

6 m

8 m

A right triangle has legs of lengths 9 cm and 12 cm. What is the length of the hypotenuse? Round to the nearest hundredth.


92 + 122 = c2 Substitute a=9 cm, b=12 cm



_____ = _____ What comes next?

______ cm = _____ How long is the hypotenuse?

Find the length of the hypotenuse of a right triangle where the legs’ lengths are a=5 cm and b=12 cm.


52 + 122 = c2 Substitute a=5 cm, b=12 cm

______ + _______ = ______ What comes

________ = _______ next?

________ = _______


a

b c

9 cm

12 cm


b

a

c

5 cm

12 cm

________ = _______ How long is the hypotenuse?

Using the same steps as the examples, find the length of the hypotenuse on each of the following right triangle to the nearest tenth. Show your work.

1.

c = _________

2.

x = _________

Now it’s your turn!

7 m

9 m c

20

12

x

3.

z = _________

Draw a picture to help you solve the following word problems. Be sure to show all work. Round to the nearest tenth.

4. A pigeon leaves its nest and flies 5 km due east. The pigeon then flies 3 km due north. How far is the pigeon from its nest?

The pigeon is __________ km from its nest.

252

500 z

5. The height of a second story window is 25 feet, and a window cleaner will need to put the ladder ten feet away from the house in order to avoid the bushes and flowers. How long of a ladder does the window cleaner need in order to achieve this task?

The ladder needs to be _________ feet long.

6. A carpenter wants to build a handicap ramp over a set of steps that is 12 feet long and 5 feet high. How long will the ramp be?

The ramp will be _________ feet long.

7. The height of the bed of a moving truck is 4 feet. The distance from the bottom edge of a ramp on the ground to the truck is 6 feet. How long is the ramp?

The ramp is ____________ long.

Solving Equations Using the Pythagorean Theorem When you draw a diagonal of a rectangle, you separate the rectangle into two right triangles.

hypotenuse

legs diagonal

8. The walls of the Downtown Recreation Center are being covered with paneling. The doorway into one room is 0.9 meters wide and 2.5 meters high. What is the length of the longest rectangular panel that can be taken through the doorway diagonally?

9. In a baseball diamond, the distance between each of the three bases and home plate are 90 feet and all form right angles. How far does the second baseman have to throw the ball in order to get the runner out before he slides into home plate?

10. Television sets are generally measured diagonally, thus classifying them as 13 inches, 27 inches, 36 inches, and so forth. You want to purchase an entertainment center, but it only has enough room in its cubicle for a 27 inch TV set. You know that the length of your TV is 15 inches, and the height of our TV is 12 inches. Will your TV be able to fit into the cubicle?

a2 + b2 = c2

92 + x2 = 152

81 + x2 = 225

81 – 81 + x2 = 225 – 81

x2 = 144

!

x 2 =

!

144 x = 12

12 in.

What if the missing side is NOT the hypotenuse? The theorem still works.

X 15

9

EXAMPLE – Finding the Leg

Find the length of the missing leg to the tenth.

11.

12.

13. A right triangle has a 47-inch hypotenuse and a 19-inch leg. What is the length of the other leg?

10 y

2.5

x

9

12.7

Use the triangle at the right. Find the length of the missing side to the nearest tenth.

1. a = 6, b = 8, c = ______ 2. a = 8, b = 24, c = ______

3. a = 1, b = 2, c = ______ 4. a = 3, b = ______, c = 5

5. a = ______, b = 7, c = 10 6. a = ______, b = 12, c = 13

7. a = 4, b = 4, c = ______ 8. a = 13, b = 2, c = ______

9. a = 15, b = ______, c = 25 10. a = ______, b = 10, c = 15

11. a = 75, b = ______, c = 100 12. a = ______, b = 12, c = 18

13. How far up a wall will an 11 m ladder reach, if the foot of the ladder must be 4 m from the base of the wall?

Check your work on problems # 1 – 13 with the answer key at the end of this packet. Then complete the following set of questions as homework for review.

Homework Problems

c

a

b

14. Frank Rd. and James Rd. make a perpendicular intersection. The state wants to build a new road. The new road will intersect 3 miles north of the intersection on Frank Rd. and 4 miles west of the intersection of James Rd. (a) How long will the new road be that intersects Frank and James roads?

(b) The new road would cost $10 per foot to pave. What would be the cost of the new road?

James Rd.

Fran

k R

d.

15. (a) A gondola travels between two elevations along a cable. What is the distance the gondola travels from the bottom of the hill to the top?

(b) The gondola travels from the bottom of the hill to the top of the hill in 20 minutes. What is the average speed of the gondola in feet per minute?

Upper elevation 7761 ft.

Lower elevation 6421 ft.

3350 ft.

16. (a) Find the length of the glass insert for the solar heating panel shown. Round your answer to the nearest tenth.

(b) What is the area of the rectangular surface of the solar panel?

15 in.

21 in.

x

24 in.

Example 2 10 m = c

Example 3

!

255 = c 2 15. 97 cm = c

Example 4 25 + 144 = c2

169 = c2

13 cm = c

1. c = 11.4 m 2. x = 23.32 3. z = 559.9 4. 5.8 km 5. 26.9 ft. 6. 13 ft. 7. 7.2 ft. 8. 2.7 m 9. 127.3 ft. 10. Yes, the TV can fit. The TV set is 19.2 in. diagonally. 11. y = 9.7 12. x = 9.0 13. 43.0 in.

ANSWERS to Examples 2 – 4 and problems # 1 - 13

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513 Pythagorean Worked Examples