1. Using the Pythagorean Theorem
Lesson 3.3.2

2. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
California Standards:
Measurement and Geometry 3.2
Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections.
Measurement and Geometry 3.3
Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement.
What it means for you:
You’ll see how to use the Pythagorean theorem to find missing side lengths of right triangles.
Key words:
Pythagorean theorem
right triangle
hypotenuse
legs
square root

3. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
In the last Lesson, you met the Pythagorean theorem and saw how it linked the lengths of the sides of a right triangle.
Area = c2
Area = b2 c b a
Area = a2
In this Lesson, you’ll practice using the theorem to work out missing side lengths in right triangles.

4. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
Use the Pythagorean Theorem to Find the Hypotenuse b a c
If you know the lengths of the two legs of a right triangle you can use them to find the length of the hypotenuse.
The theorem says that c2 = a2 + b2 where c is the length of the hypotenuse, and a and b are the lengths of the two legs.
So if you know the lengths of the legs you can put them into the equation, and solve it to find the length of the hypotenuse.

5. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
Example 1
Use the Pythagorean theorem to find the length of the hypotenuse of the right triangle shown.
c cm
8 cm
Solution
c2 = a2 + b2
First write out the equation
6 cm
c2 =
Substitute in the side lengths that you know
c2 =
Simplify the expression
c2 = 100
c =
Take the square root of both sides
c = 10 cm
Solution follows…

6. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
A lot of the time your solution won’t be a whole number.
That’s because the last step of the work is taking a square root, which often leaves a decimal or an irrational number answer.
c2 = a2 + b2
c2 =
c2 =
c2 = 100
c =
c = 10 cm

7. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
Example 2
Use the Pythagorean theorem to find the length of the hypotenuse of the right triangle shown.
c cm
1 m
Solution
c2 = a2 + b2
First write out the equation
1 m
c2 =
Substitute in the side lengths that you know
c2 = 1 + 1
Simplify the expression
c2 = 2
Cancel out the squaring by taking out the square root
c = m
If you do this calculation on a calculator, you’ll see that m is approximately equal to 1.4 m.
Solution follows…

8. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem 1 2 3 4 5 y x A B
The Pythagorean theorem is also useful for finding lengths on graphs that aren’t horizontal or vertical.

9. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
Example 3 1 2 3 4 5 y x K L
Find the length of the line segment KL.
Solution
3 units
Draw a horizontal and vertical line on the plane to make a right triangle.
2 units
Now use the same method as before.
Solution continues…
Solution follows…

10. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
Example 3 1 2 3 4 5 y x K L
Find the length of the line segment KL.
Solution (continued)
3 units
KL2 = a2 + b2
Write out the equation
KL2 =
Substitute in the side lengths that you know
2 units
KL2 = 9 + 4
Simplify the expression
KL2 = 13
KL =
Cancel out the squaring by taking the square root
KL » 3.6 units

11. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
Guided Practice
Use the Pythagorean theorem to find the length of the hypotenuse in Exercises 1–3. 1. 2. 3.
15 units
8 units
3.6 cm
c cm
c ft
12 cm
c units
1.5 cm
5 ft
c2 = c2 = c2 = 169 c = 13 ft
c2 = c2 = c2 = 289 c = 17 units
c2 = c2 = c2 = c = 3.9 cm
Solution follows…

12. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
Guided Practice
4. Use the Pythagorean theorem to find the length of the line segment XY. y x –1 –2 1 2 3 X Y
XY2 = XY2 = XY2 = XY2 = XY » 4.2 units
Solution follows…

13. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
You Can Use the Theorem to Find a Leg Length
If you know the length of the hypotenuse and one of the legs, you can use the theorem to find the length of the “missing” leg.
You just need to rearrange the formula:
a2 + b2 = c2
Subtract b2 from both sides
to get the a2 term by itself.
a2 = c2 – b2
Remember that it doesn’t matter which of the legs you call a and which you call b. But the hypotenuse is always c.
Now you can substitute in values to find the missing leg length as you did with the hypotenuse.

14. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
Example 4
Find the missing leg length in this right triangle. cm
Solution
3 cm
c2 = a2 + b2
First write out the equation a
a2 = c2 – b2
Rearrange it
a2 = – 32
Substitute in the side lengths that you know
a2 = 58 – 9
Simplify the expression
a2 = 49
a =
Take the square root of both sides
a = 7 cm
Solution follows…

15. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
Guided Practice
Use the Pythagorean theorem to calculate the missing leg lengths in Exercises 5–8. 5. 6.
a ft
1.6 ft
20 cm
16 cm
a2 = 202 – 162 a2 = 400 – 256 a2 = 144 a = 12 cm
3.4 ft
a2 = 3.42 – 1.62 a2 = – 2.56 a2 = 9 a = 3 ft
a cm 7.
a units 8.
a units
10 units
units
a2 = 136 – 102 a2 = 136 – 100 a2 = 36 a = 6 units
a2 = 89 – 52 a2 = 89 – 25 a2 = 64 a = 8 units
units
5 units
Solution follows…

16. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
Independent Practice
Use the Pythagorean theorem to find the value of c in Exercises 1–5. 1. 2. 3.
c cm
c m
12 cm
0.8 m
4.8 m
3.6 m
c = 15
c = 1
c m
0.6 m
9 cm
c = 6 4.
1 cm 5.
c in
1.5 cm
c cm
7 in
c =
c =
2 in
Solution follows…

17. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
Independent Practice
Calculate the value of a in Exercises 6–10. 6. 7. 8.
a cm
a = 0.9
7.5 m
a m
5 feet
4 feet
a = 6
4 cm
4.1 cm
4.5 m
a feet
a = 3 9.
10.
units
3 units
3 in
a in
a =
a units
a = 6 in
Solution follows…

18. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
Independent Practice
11. Find the length of line AB.
12. Find the perimeter of quadrilateral ABCD 1 2 3 4 5 y x A B B y –1 –2 1 2 3 A x C
» 5.1 units D
» 12.8 units
Solution follows…

19. Using the Pythagorean Theorem
Lesson
3.3.2
Using the Pythagorean Theorem
Round Up
The Pythagorean theorem is really useful for finding missing side lengths of right triangles.
If you know the lengths of both legs of a triangle, you can use the formula to work out the length of the hypotenuse.
And if you know the lengths of the hypotenuse and one of the legs, you can rearrange the formula and use it to work out the length of the other leg.