Free Pre-Algebra
Lesson 50 ! page 1
Lesson 50
The Pythagorean Theorem
The Pythagorean Theorem is one of the most famous results in geometry. From the ancient Babylonians to the scarecrow in
the Wizard of Oz and U.S. President Garfield, the theorem has been part of world history. Although in western mathematics
the theorem named for Pythagoras, a Greek, it was probably independently discovered in both India and China before
Pythagoras. There are literally hundreds of proofs of the theorem. You can read a bit more about the history here.
What is a Theorem?
A theorem in mathematics is a statement about mathematical
relationships that can be proved. If you’ve taken a geometry
course, you may remember doing proofs. If we make some
assumptions about geometry that we all agree about, and then
draw logical conclusions from those assumptions, we agree on
the conclusions as well. This is called proving a theorem. A
theorem in mathematics is not like a theory in science. Scientists
keep challenging and improving their theories to explain realworld data. Once a theorem has been proved in mathematics, it
is no longer in question.
What is the Pythagorean Theorem?
The Pythagorean theorem states the relationship between the
three sides of a right triangle. (A right triangle has a square
corner.)
Pythagoras
Pythagoras lived more than
2,500 years ago in ancient
Greece, so it’s hard to be really
definite about him.
He was a teacher and the
leader of a “mystery,” a
religious cult. The cult saw
number as the foundation of the world, evident in
musical harmonies and the movement of the stars.
The Pythagorean mysteries profoundly influenced the
Greek philosopher Plato.
Pythagoras was quite famous and influential in his
own time, so much so that he was considered a
political threat and was persecuted and killed. But
much of the information we have now about him is
considered unreliable by many scholars.
In a right triangle, the two sides around the right angle (shown with
the little square corner) are called the legs. The longer side, across
from the right angle, is called the hypotenuse.
The legs are usually labeled a and b. It doesn’t matter which is
which. The hypotenuse is labeled c. It is important which side is the
hypotenuse. You can find it by thinking of the square that indicates
a right angle as an arrowhead, pointing to the right angle.
There are squares drawn on each side of the triangle, because the
relationship between the sides is really a relationship between the
areas of these squares. The two smaller squares, the ones on the
legs, add up to equal in area the larger square, the one on the hypotenuse. Written in algebraic notation,
a2 + b2 = c2
!----------- That’s it. The Pythagorean Theorem
It’s a formula that relates the three sides of a right triangle.
In the triangle to the right, we see that the formula would read: 32 + 4 2 = 52 ,
9 + 16 = 25 , which is true.
You can see a flash animation of a proof of the theorem at
http://themetapicture.com/media/why-couldnt-i-have-been-shown-this-in-maths-class.gif.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 50 ! page 2
Solving Triangles
We use the Pythagorean Theorem like any other formula, to find one of the numbers when we know all the others. To find
the side of the triangle we need to take a square root.
Example: Find the length of side c.
The sides are labeled with the same letters as the formula, so we don’t need to figure
out which side is the hypotenuse. Just use the sides a and b with the formula.
a2 + b2 = c2
(5) + (12)
2
2
= 25 + 144 = 169
We know c is a positive number, since it measures a length. That means we
can write the related square and square root equations with c. We know c2 is
169, so we can take the square root of 169 to find c.
Side c measures 13 cm.
c 2 = 169
c 2 = 169
169 = c
169 = 13
Example: Find the length of the leg of the right triangle.
The sides are labeled with the same letters as the formula, so we don’t need to figure
out which side is the hypotenuse. You will need to solve an equation, though.
a2 + b2 = c2
( ) = ( 41)
a 2 + 40
2
2
a 2 + 1600 = 1681
a 2 = 1681! 1600 = 81
Again, since the sides have positive lengths, to find the side when we know the
square, we take the square root.
Side a measures 9 cm.
a 2 = 81
81 = c
81 = 9
Alternate Forms of the Pythagorean Theorem
The formula a 2 + b 2 = c 2 can be re-written so that we
can find any of the sides without solving an equation. You
can use these alternate forms if you prefer.
a = c2 ! b2
b = c2 ! a2
c = a2 + b2
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 50 ! page 3
Example: Find the missing length in the right triangle.
Here the sides are not labeled, so it is up to you to determine which side is the
hypotenuse. Look for the right angle in the corner, then use it to point straight across
at the hypotenuse, side c. That is the side we are looking for.
The other two sides are a and b, and it doesn’t matter which is which. The remaining
steps are the same.
a2 + b2 = c2
(30) + (16)
2
2
= c2
900 + 256 = c 2
Take the square root to find the length of the missing side.
c 2 = 1156
c 2 = 1156
The hypotenuse measures 34 cm.
1156 = c
1156 = 34
At this point you may be wondering if all right triangles work out so beautifully, with all three sides having whole number
measurements. When three whole numbers satisfy the formula a 2 + b 2 = c 2 , they are called a Pythagorean Triple. For
example, the numbers 3, 4, and 5 are a Pythagorean triple, because 32 + 42 = 52. Lists of Pythagorean triples go back as far
as Babylonian times. Here’s one online: http://www.tsm-resources.com/alists/trip.html. Right triangles with all three sides
having whole number lengths are special.
Here’s a right triangle whose sides do not form a Pythagorean triple:
Example: Find the missing length in the right triangle.
The missing length is the hypotenuse, side c in the formula.
a2 + b2 = c2
(1) + (1)
2
2
= c2
1+ 1= c 2
Take the square root to find the length of the missing side.
The hypotenuse measures about 1.4 cm.
c2 = 2
c2 = 2
2 =c
2 ! 1.414
In any right triangle, the hypotenuse is the longest side. You can use that fact as a common sense check on your answers.
If a leg works out to be longer than the hypotenuse, check your work! It doesn’t matter which way a right triangle is oriented,
the hypotenuse is always across from the right angle.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 50 ! page 4
Example: Find the missing length in the right triangle.
The missing length is one of the legs. We can call it either a or b. The side across from the right
angle, measuring 9 cm, is the hypotenuse.
a2 + b2 = c2
()
2
a 2 + 8 = 92
a 2 + 64 = 81
Take the square root to find the length of the missing side.
The leg measures about 4.1 cm.
a 2 = 81! 64 = 17
a 2 = 17
17 = a
17 ! 4.123
!
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 50 ! page 5
Lesson 50: The Pythagorean Theorem
Worksheet
Name ________________________________________
1. Label the sides of the triangle a, b, and c. Label the legs
and hypotenuse.
2. Label the sides of the triangle a, b, and c. Label the legs
and hypotenuse.
3. Find the length of the hypotenuse.
4. Find the length of the leg.
5. Find the missing length.
6. Find the missing length.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Challenge
Draw the square root spiral. Use the starter triangle and follow the steps.
Step 1
Step 2
Step 3
The first right triangle has
On the hypotenuse of the
Connect the endpoint to
both legs 1 cm long.
first triangle, draw a right
make another triangle.
angle and make a side 1 cm
long.
Lesson 50 ! page 6
Repeat
Repeat Steps 2 and 3 on the
new triangle.
and continue.
Why is this called the square root spiral? Find the lengths of the hypotenuses of the first few triangles, the explain.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 50 ! page 7
Lesson 50: The Pythagorean Theorem
Homework 50A
Name________________________________________
1. Solve the equation 5 y ! 9 = 41.
2. Solve the equation 6 ! 7x = !55
3. Use the structure sentence to write an equation and solve.
4. A rectangle has perimeter 96 meters. The length is 2
meters. What is the width?
A box of candy had 15 nut chews with 85 calories each and
25 carmels. If the total number of calories in the box is 3525,
how many calories in a caramel?
number of candies • calories per candy +
number of candies • calories per candy = total calories
5. Simplify the fraction.
6. Multiply the fractions.
48a 2b 2c
60ab 3c 2
7x 4 y 2
•
2 y 21
7. Find equivalent fractions with a common denominator.
8. Combine like terms.
8
25a
8a + 5a + 65 ! 24 ! 3a
13
5a 2
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 50 ! page 8
(
)
(
) (
)
9. Use the distributive property to simplify !9 3x ! 4 .
10. Simplify !5 7x ! 12 + 6 9x + 1 .
11. Evaluate. Round to three decimal places if rounding is
necessary.
12. Evaluate. Round to three decimal places if rounding is
necessary.
a.
121
b.
!121
c. ! 121
d.
(!11)2
a.
100
25
c.
36 • 9
d.
36 • 9
b.
100
25
13. Write the related square and square root problems for
(1/2)2 and (1/3)2.
14. Use the Pythagorean Theorem to find the missing side.
Round to one decimal place if rounding is necessary.
15. Use the Pythagorean Theorem to find the missing side.
Round to one decimal place if rounding is necessary.
16. Use the Pythagorean Theorem to find the missing side.
Round to one decimal place if rounding is necessary.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 50 ! page 9
Lesson 50: The Pythagorean Theorem
Homework 50A Answers
2. Solve the equation 6 ! 7x = !55
1. Solve the equation 5 y ! 9 = 41.
5 y ! 9 = 41
5 y = 50
5 y ! 9 + 9 = 41+ 9
5 y / 5 = 50 / 5
y = 10
3. Use the structure sentence to write an equation and solve.
A box of candy had 15 nut chews with 85 calories each and
25 carmels. If the total number of calories in the box is 3525,
how many calories in a caramel?
number of candies • calories per candy +
number of candies • calories per candy = total calories
15 • 85 + 25 • c = 3525
25c + 1275 = 3525
25c + 1275 ! 1275 = 3525 ! 1275
25c = 2250
25c / 25 = 2250 / 25
c = 90
!7x + 6 = !55
!7x = !61
61
x=
7
! 7x + 6 ! 6 = !55 ! 6
! 7x / (!7) = !61/ (!7)
4. A rectangle has perimeter 96 meters. The length is 2
meters. What is the width?
P = 2L + 2W
96 = 2(2) + 2W
2W + 4 = 96
2W + 4 ! 4 = 96 ! 4
2W = 92
2W / 2 = 92 / 2
W = 46
The width is 46 meters.
There are 90 calories per caramel.
5. Simplify the fraction.
6. Multiply the fractions.
48a 2b 2c
7x 4 y 2
•
2 y 21
60ab 3c 2
=
2 • 2 •2•2• 3 • a •a • b • b • c
2 • 2 • 3 •5• a • b • b •b • c •c
=
4a
5bc
=
7 •x
2• y
•
2 •2• y • y
3• 7
7. Find equivalent fractions with a common denominator.
8. Combine like terms.
8
a
8a
8
=
• =
25a 5 • 5 • a a 25a 2
8a + 5a + 65 ! 24 ! 3a
13
13
5
65
=
• =
2
5 • a • a 5 25a 2
5a
© 2010 Cheryl Wilcox
=
2xy
3
= (8a + 5a ! 3a ) + (65 ! 24)
= 10a + 41
Free Pre-Algebra
Lesson 50 ! page 10
(
)
9. Use the distributive property to simplify !9 3x ! 4 .
(
11. Evaluate. Round to three decimal places if rounding is
necessary.
121 = 11
b.
!121 not a real number
12. Evaluate. Round to three decimal places if rounding is
necessary.
a.
c. ! 121 = !11
d.
(!11)2 = 121 = 11
13. Write the related square and square root problems for
(1/2)2 and (1/3)2.
2
)
= !5(7x ) + !5(!12) + 6(9x ) + 6(1)
= !35x + 60 + 54x + 6
= 19x + 66
= !9(3x ) + !9(!4)
= !27x + 36
a.
) (
10. Simplify !5 7x ! 12 + 6 9x + 1 .
100
25
=
10
=2
5
100
= 4 =2
25
b.
c.
36 • 9 = 324 = 18
d.
36 • 9 = 6 • 3 = 18
14. Use the Pythagorean Theorem to find the missing side.
Round to one decimal place if rounding is necessary.
2
! 1$
1
#" 2 &% = 4
! 1$
1
#" 3 &% = 9
1 1
=
4 2
1 1
=
9 3
a 2 + b 2 = c 2 , missing side is c
202 + 212 = c 2
400 + 441= c 2
c 2 = 841
c = 841 = 29
The missing side c is 29 cm.
15. Use the Pythagorean Theorem to find the missing side.
Round to one decimal place if rounding is necessary.
a2 + b2 = c2
a 2 + b 2 = c 2 , missing side is a
missing side is c
19 + 9 = c
2
2
2
c = 442
2
361+ 81= c
2
c = 442 ! 21.024
The missing side c is about 21.0 cm.
© 2010 Cheryl Wilcox
16. Use the Pythagorean Theorem to find the missing side.
Round to one decimal place if rounding is necessary.
a 2 + 122 = 152
a 2 + 144 = 225
a 2 = 225 ! 144 = 81
a = 81 = 9
The missing side a is 9 cm.
Free Pre-Algebra
Lesson 50 ! page 11
Lesson 50: The Pythagorean Theorem
Homework 50B
Name_________________________________________
1. Solve the equation 12x + 17 = !7 .
2. Solve the equation 3 ! 4x = 1.
3. Use the structure sentence to write an equation and solve.
4. A rectangle has perimeter 54 meters. The length is 8
meters. What is the width?
A bag of birthday favors had 8 kazoos, which cost $1.29
each, and 16 candy whistles. The total cost of the bag is
$24.56. What does a candy whistle cost.
number of kazoos • price per kazoo +
number of whistles • price per whistle = total cost
5. Simplify the fraction.
6. Multiply the fractions.
54x 3 y 2z
12m 14n 2
•
35n
9
90xy 2
7. Find equivalent fractions with a common denominator.
8. Combine like terms.
3
5x
5w ! 9g ! 9w + 5g + 19
1
x
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 50 ! page 12
(
9. Use the distributive property to simplify !11 !4x ! 4
)
(
)
(
)
10. Simplify 4 11x ! 5 + !4 10x ! 6 .
.
11. Evaluate. Round to three decimal places if rounding is
necessary.
a. ! 196
b.
c.
(
!14
)
12. Evaluate. Round to three decimal places if rounding is
necessary.
a.
2
( !14 )
144
16
c.
16 • 25
d.
16 • 25
2
b.
144
16
13. Write the related square and square root problems for
(1/4)2 and (1/5)2.
14. Use the Pythagorean Theorem to find the missing side.
Round to one decimal place if rounding is necessary.
15. Use the Pythagorean Theorem to find the missing side.
Round to one decimal place if rounding is necessary.
16. Use the Pythagorean Theorem to find the missing side.
Round to one decimal place if rounding is necessary.
© 2010 Cheryl Wilcox