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Free Pre-Algebra Lesson 51 ! page 1

© 2010 Cheryl Wilcox

Lesson 51

Practical Uses of the Pythagorean Theorem

The Pythagorean Theorem is primarily used in architecture, construction, surveying, and engineering.

“Modern carpentry work is so much easier when the Pythagorean Theorem is applied to the task at hand. Roof framing, squaring walls, and foundations rely on this basic principle of mathematics.”

Carpentry-Pro-Framer http://www.carpentry-pro-framer.com/pythagorean-theorem.html

Builders often need to construct a square corner. The Pythagorean Theorem tells us that if a corner is square, then the

sides of a triangle built on that corner will satisfy the formula a2 + b2 = c2. But we can also prove that the converse of the

theorem is true. If the sides of a triangle satisfy the formula a2 + b2 = c2, then the triangle is a right triangle, with a square corner. (The converse switches the “if” “then” clauses of the sentence.) This fact has been used since ancient times to construct square corners.

To make a square corner, the ancient Egyptians used a rope marked with

twelve even segments. That’s because

twelve segments can make a triangle with

sides 3, 4, and 5. That triangle satisfies the formula a2 + b2 = c2 (32 + 42 = 52), and so a 3-4-5 triangle has a right angle. The rope-stretchers would stake out the triangle with the rope and so

mark a right angle.

Example: If the framing shown is constructed correctly with a right angle in the corner, what will the carpenter’s tape measure read on the diagonal?

The diagonal is the hypotenuse of a right triangle with legs 15 inches and 36 inches. According to the Pythagorean Theorem,

a2+b

2= c

2

152+ 36

2= 225 + 1296 = 1521

c = 1521 = 39

The tape measure should read 39 inches.

Example: What length should the wires be to hold the pole at a right angle to the roof?

There are two wires, each the hypotenuse of a separate right triangle.

a2+b

2= c

2

402+ 90

2= 9700

c = 9700 = 98.489

a2+b

2= c

2

202+ 90

2= 8500

c = 8500 = 92.195

One wire should be about 98.5 inches and the other about 92.2 inches.



Example: A carpenter is using a 20 foot ladder against a straight wall. The top of the ladder rests against the wall. The ANSI/OHSA safety regulations for ladders instruct: “Make sure the ladder is

about 1 foot away from the vertical support for every 4 feet of ladder height

between the foot and the top support.”

a) How far away from the wall should she place the foot of the ladder?

If the ladder is 1 foot away for every 4 feet of ladder length, the ratio of distance from wall to ladder length is 1:4.

1 ft

4 ft=

x ft

20 ft, and cross-multiplying, 4x = 20, so x = 5 ft.

b) How far up the wall will the ladder reach?

On the figure shown, we know that A (ladder length) = 20 feet. The ladder length is the

hypotenuse of the right triangle the ladder makes with the wall, so side c = 20. One leg of the triangle is 5 feet, so we’ll call that leg b and find the length of the other leg,

labeled h in the diagram.

a2+b

2= c

2

h2+ 5

2= 20

2

h2+ 25 = 400

h2= 400 ! 25 = 375

h = 375 " 18.028

The ladder will reach about 18 feet up the wall.

Example: Find length of pipe needed for the sprinkler system shown.

There are two right triangles shown:

One has legs 8 feet and 12 feet.

a2+b

2= c

2

82+ 12

2= 208

c = 208 ! 14.422

One has legs 6 feet and 8 feet.

a2+b

2= c

2

62+ 8

2= 100

c = 100 = 10

Adding all the lengths of pipe, we get

6 ft + 14.4 ft + 12 ft + 10 ft + 18 ft = 60.4 ft



Example: The pilot knows she has descended 1000 feet and that she has traveled 18,000 feet through the air. How far has she traveled in horizontal distance along the ground?

.

a2+b2

= c2

10002+b2

= 180002

1,000,000 +b2= 324,000,000

b2= 323,000,000

b = 323,000,000 ! 17,972.201

The plane has traveled about 17,972 feet

along the ground.

!



Lesson 51: Practical Uses of the Pythagorean Theorem

Worksheet Name _______________________________________

1. A carpenter measured the diagonal of a 48 inch by 60 inch gate to the nearest sixteenth of an inch and found it to

be 771/16 inches. Is the gate out of square?

2. Gutters are to be installed along the roofline and extend another 6 inches past the end of the roof. How many feet of

gutter are needed?

3. Bailey planned this garden using an online tool. The length along the bottom fence is 22 feet and the length along

the right hand fence to the end of the path at the gate is 12 feet. How long is the diagonal path?

4. A painter is using a 30 foot extension ladder. He follows the safety regulations, making sure the base of the ladder is

1 foot from the wall for every 4 feet of ladder height.

How high up the wall will the ladder reach?



5. Challenge: If you draw two lines from the ends of the diameter of a circle that meet on the edge of the circle, they

will form a right angle where they meet.

Find the circumference of the circle. (C = !d) Round to the nearest tenth.

6. Super Challenge: Can you find the length of the line marked with an x? Round to the nearest meter.




Homework 51A Name_________________________

1. Solve the equation

7

8y = 35 .


3

17n +

5

17=!3

17.

3. Simplify using the distributive property.

457

15h !

8

9

"

#$%

&'

4. Simplify by combining like terms.

4

5y +

1

5!

9

10y !

1

10

5. One box has length 5 cm, width 10 cm, and height 20 cm. The second box has double the length, width, and height.

Find the volume of each box.

.

Compare the volumes with a difference and a ratio. Write a sentence with each comparison.

6. Jon and Myra order a 16-inch diameter pizza, while TJ and Boaz order two 8-inch diameter pizzas.

Find the area of pizza each table received, rounded to the nearest square inch.

16-inch:

two 8-inch:

Compare the amounts of pizza with a difference and a ratio. Write a sentence with your comparison.



7.

8 +3 ! 5

2! 3 • 4

2

8.

8 +3 ! 5

2! 3 • 4

9. Evaluate

a. 25 • 9

b. 25 • 9

c.

81

9

d.

81

9

10. Evaluate

a. !4

b.

(!2)2

c. ! (!2)2

d.

4( )2

11. Find the length of the leg that is not marked. Round to

the nearest tenth if rounding is necessary.

13. If the diagonal measures 34 inches, is the frame square?




Homework 51A Answers


7

8y = 35 .

8

7•

7

8y = 35

5

•8

7

y = 40


3

17n +

5

17=!3

17.

3n + 5 = !3 3n + 5 ! 5 = !3 ! 5

3n = !8 3n / 3 = !8 / 3

n =!8

3


457

15h !

8

9

"

#$%

&'

= 45

3

7

15

h!

"#

$

%& + 45

5

'8

9

!

"#

$

%&

= 21h ' 40


4

5y +

1

5!

9

10y !

1

10

=4

5y !

9

10y

"

#$%

&'+

1

5!

1

10

"

#$%

&'

=8

10y !

9

10y

"

#$%

&'+

2

10!

1

10

"

#$%

&'

=!1

10y +

1

10

5. One box has length 5 cm, width 10 cm, and height 20 cm. The second box has double the length, width, and height.

Find the volume of each box.

V = lwh = (5)(10)(20) = 1000

Volume of first box is 1000 cm3

Second box has length 5•2 = 10 cm,

width 10•2 = 20 cm, and height 20•2 = 40 cm.

V = lwh = (10)(20)(40) = 8000

The volume of the second box is 8000 cm.

Compare the volumes with a difference and a ratio. Write a sentence with each comparison.

Difference: 8000 – 1000 = 7000 cm3

The second box with double the sides is

7000 cubic cm larger than the first box.

Ratio: 8000 / 1000 = 8.

The second box with double the sides is eight

times as large as the first box.

6. Jon and Myra order a 16-inch diameter pizza, while TJ and Boaz order two 8-inch diameter pizzas.

Find the area of pizza each table received, rounded to the nearest square inch.

16-inch: A = !r 2= !(16)2

" 804 in2

two 8-inch: 2A = 2!r 2= 2!(8)2

" 402 in2

Compare the amounts of pizza with a difference and a ratio.

Write a sentence with your comparison.

Difference: 804 – 402 = 402 in2

The 16-inch pizza is 402 sq in greater in area

than the two 8-inch pizzas put together.

Ratio: 804/402 = 2

The 16-inch pizza is twice as big as

the two 8-inch pizzas.



7.

8 +3 ! 5

2! 3 • 4

2

= 8 +!2

2! 3 • 4

2

= 8 +!2

2! 3 • 16

= 8 + !1! 3 • 2

= 8 + !1! 6

= 1

8.

8 +3 ! 5

2! 3 • 4

= 8 +!2

2! 3 • 4

= 8 +!2

2! 3 • 2

= 8 + !1! 3 • 2

= 8 + !1! 6

= 1

9. Evaluate

a. 25 • 9 = 5 • 3 = 15

b. 25 • 9 = 225 = 15

c.

81

9

=9

3= 3

d.

81

9 = 9 = 3

10. Evaluate

a. !4 not a real number

b.

(!2)2

= 4 = 2

c. ! (!2)2

= ! 4 = !2

d.

4( )2

= 22= 4

11. Find the length of the leg that is not marked. Round to


a2+b2

= c2

(11.1)2+b2

= (19.2)2

123.21+b2= 368.64

b2= 368 ! 123.21= 245.43

b = 245.43 " 15.666

The leg is about 15.7 cm.

13. If the diagonal measures 34 inches, is the frame square?

a2+b

2= c

2

302+ 16

2= c

2

900 + 256 = c2

c2= 1156

c = 1156 ! 34

Yes, since 302 + 162 = 342, the corner

must be square (a right angle).




Homework 51B Name______________________________________


9

10m = 108 .


7

16n !

9

16=!1

16.


727

8t !

8

9

"

#$%

&'


3x

2+

5

4!

5x

4!

3

2

5. One box has length 2 cm, width 5 cm, and height 10 cm. The second box has length, width, and height each 1 cm

more than the first box. Find the volume of each box.

Compare the volumes with a difference and a ratio. Write a

sentence with each comparison.

6. Eric and Kandy ordered a round cake with diameter 9 inches. Laura and Javier ordered an 8-inch diameter cake.

Both cakes are the same height.

Find the area of cake each couple ordered, rounded to the

nearest square inch.

9-inch:

t8-inch:

Compare the cakes with a difference and a ratio. Write a

sentence with your comparison.



7.

!4 +6

2

2! 10

8.

!4 +36

2! 10

9. Evaluate

a. 36 • 4

b. 36 • 4

c.

144

9

d.

144

9

10. Evaluate

a.

(!7)(7)

b.

(!7)2

c.

49( )2

d. ! 49( )

2

11. Find the length of the side that is not marked. Round to


m.

13. How much distance do you save by walking across the

grass instead of taking the path?

lesson 51 text - diablo valley collegevoyager.dvc.edu/~lmonth/prealg/lesson51student.pdf · the...

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