Free Pre-Algebra
Lesson 51 ! page 1
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 + b2 = c2
152 + 362 = 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 + b2 = c2
a2 + b2 = c2
402 + 902 = 9700
202 + 902 = 8500
c = 9700 = 98.489
c = 8500 = 92.195
One wire should be about 98.5 inches and the other about 92.2 inches.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 51 ! page 2
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
x ft
=
, and cross-multiplying, 4x = 20, so x = 5 ft.
4 ft 20 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 + b2 = c2
h 2 + 52 = 202
h 2 + 25 = 400
h 2 = 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.
One has legs 6 feet and 8 feet.
a +b =c
a2 + b2 = c2
2
2
82 + 122 = 208
62 + 82 = 100
c = 208 ! 14.422
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
© 2010 Cheryl Wilcox
2
Free Pre-Algebra
Lesson 51 ! page 3
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 + b 2 = 180002
2
. 1,000,000 + b = 324,000,000
b 2 = 323,000,000
b = 323,000,000 ! 17,972.201
The plane has traveled about 17,972 feet
along the ground.
!
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 51 ! page 4
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.
© 2010 Cheryl Wilcox
How high up the wall will the ladder reach?
Free Pre-Algebra
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.
© 2010 Cheryl Wilcox
Lesson 51 ! page 5
6. Super Challenge: Can you find the length of the line
marked with an x? Round to the nearest meter.
Free Pre-Algebra
Lesson 51 ! page 6
Lesson 51: Practical Uses of the Pythagorean Theorem
Homework 51A
1. Solve the equation
7
y = 35 .
8
Name_________________________
2. Solve the equation
3
5
!3
n+
=
.
17
17 17
3. Simplify using the distributive property.
4. Simplify by combining like terms.
" 7
8%
45 $ h ! '
9&
# 15
4
1 9
1
y+ !
y!
5
5 10
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.
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 volumes with a difference and a ratio. Write a
sentence with each comparison.
© 2010 Cheryl Wilcox
Compare the amounts of pizza with a difference and a ratio.
Write a sentence with your comparison.
Free Pre-Algebra
7. 8 +
3!5
! 3 • 42
2
9. Evaluate
Lesson 51 ! page 7
8. 8 +
3!5
!3• 4
2
10. Evaluate
a.
25 • 9
a.
!4
b.
25 • 9
b.
(!2)2
81
c. ! (!2)
c.
d.
9
81
9
11. Find the length of the leg that is not marked. Round to
the nearest tenth if rounding is necessary.
© 2010 Cheryl Wilcox
2
d.
( )
4
2
13. If the diagonal measures 34 inches, is the frame square?
Free Pre-Algebra
Lesson 51 ! page 8
Lesson 51: Practical Uses of the Pythagorean Theorem
Homework 51A Answers
1. Solve the equation
7
y = 35 .
8
5
8 7
8
• y = 35 •
7 8
7
y = 40
2. Solve the equation
3
5
!3
n+
=
.
17
17 17
3n + 5 = !3
3n = !8
!8
n=
3
3n + 5 ! 5 = !3 ! 5
3n / 3 = !8 / 3
3. Simplify using the distributive property.
4. Simplify by combining like terms.
" 7
8%
45 $ h ! '
9&
# 15
4
1 9
1
y+ !
y!
5
5 10
10
5
! 7 $
! 8$
= 45 #
h & + 45 # ' &
" 15 %
" 9%
3
= 21h ' 40
"4
9 % "1 1%
=$ y!
y +
!
10 '& $# 5 10 '&
#5
" 8
9 % " 2
1%
=$
y!
y' + $
!
10 & # 10 10 '&
# 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
!1
1
y+
10
10
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 = ! (16) " 804 in
2
2
2
two 8-inch: 2A = 2!r = 2! (8) " 402 in
2
2
2
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.
© 2010 Cheryl Wilcox
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.
Free Pre-Algebra
7. 8 +
Lesson 51 ! page 9
3!5
! 3 • 42
2
8. 8 +
3!5
!3• 4
2
!2
! 3 • 42
2
!2
= 8+
! 3 • 16
2
= 8 + !1! 3 • 2
= 8 + !1! 6
=1
!2
!3• 4
2
!2
= 8+
!3•2
2
= 8 + !1! 3 • 2
= 8 + !1! 6
=1
= 8+
9. Evaluate
= 8+
10. Evaluate
a.
25 • 9 = 5 • 3 = 15
a.
!4 not a real number
b.
25 • 9 = 225 = 15
b.
(!2)2 = 4 = 2
c.
d.
81
9
=
9
=3
3
81
= 9=3
9
11. Find the length of the leg that is not marked. Round to
the nearest tenth if rounding is necessary.
c. ! (!2) = ! 4 = !2
2
d.
( ) =2
4
2
2
=4
13. If the diagonal measures 34 inches, is the frame square?
a2 + b2 = c2
(11.1)2 + b 2 = (19.2)2
a2 + b2 = c2
123.21+ b 2 = 368.64
302 + 162 = c 2
b 2 = 368 ! 123.21= 245.43
b = 245.43 " 15.666
The leg is about 15.7 cm.
900 + 256 = c 2
c 2 = 1156
c = 1156 ! 34
Yes, since 302 + 162 = 342, the corner
must be square (a right angle).
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 51 ! page 10
Lesson 51: Practical Uses of the Pythagorean Theorem
Homework 51B
1. Solve the equation
9
m = 108 .
10
Name______________________________________
2. Solve the equation
7
9
!1
n!
=
.
16
16 16
3. Simplify using the distributive property.
4. Simplify by combining like terms.
"7
8%
72 $ t ! '
9&
#8
3x 5 5x 3
+ !
!
2 4 4 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.
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 volumes with a difference and a ratio. Write a
sentence with each comparison.
© 2010 Cheryl Wilcox
Compare the cakes with a difference and a ratio. Write a
sentence with your comparison.
Free Pre-Algebra
7. !4 +
Lesson 51 ! page 11
62
! 10
2
36
! 10
2
8. !4 +
9. Evaluate
10. Evaluate
a.
36 • 4
a.
(!7)(7)
b.
36 • 4
b.
(!7)2
c.
d.
( )
d. ( ! 49 )
144
c.
9
2
2
144
9
11. Find the length of the side that is not marked. Round to
the nearest tenth if rounding is necessary.
m.
© 2010 Cheryl Wilcox
49
13. How much distance do you save by walking across the
grass instead of taking the path?