Lesson 3.3.2Lesson 3.3.2

Using the Pythagorean Theorem

Using the Pythagorean Theorem

2

Lesson

3.3.2

California Standards:Measurement and Geometry 3.2Understand 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 3.3Know 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.

Using the Pythagorean TheoremUsing the Pythagorean Theorem

Key words:• Pythagorean theorem• right triangle• hypotenuse• legs• square root

3

Lesson

3.3.2

In the last Lesson, you met the Pythagorean theorem and saw how it linked the lengths of the sides of a right triangle.

In this Lesson, you’ll practice using the theorem to work out missing side lengths in right triangles.

Using the Pythagorean TheoremUsing the Pythagorean Theorem

b

a

c

Area = c2

Area = b2

Area = a2

4

Lesson

3.3.2

Use the Pythagorean Theorem to Find the Hypotenuse

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.

Using the Pythagorean TheoremUsing the Pythagorean Theorem

b

a

c

5

Lesson

3.3.2

Use the Pythagorean theorem to find the length of the hypotenuse of the right triangle shown.

Example 1

Solution follows…

Using the Pythagorean TheoremUsing the Pythagorean Theorem

c cm

6 cm

8 cm

Solution

c2 = a2 + b2

c2 = 62 + 82

First write out the equation

Substitute in the side lengths that you know

Simplify the expression

Take the square root of both sides

c2 = 36 + 64

c2 = 100

c =

c = 10 cm

6

Lesson

3.3.2

A lot of the time your solution won’t be a whole number.

Using the Pythagorean TheoremUsing the Pythagorean Theorem

c2 = a2 + b2

c2 = 62 + 82

c2 = 36 + 64

c2 = 100

c =

c = 10 cm

That’s because the last step of the work is taking a square root, which often leaves a decimal or an irrational number answer.

7

Lesson

3.3.2

Use the Pythagorean theorem to find the length of the hypotenuse of the right triangle shown.

Example 2

Solution follows…

Using the Pythagorean TheoremUsing the Pythagorean Theorem

c cm

1 m

1 m

Solution

c2 = a2 + b2

c2 = 12 + 12

First write out the equation

Substitute in the side lengths that you know

Simplify the expression

Cancel out the squaring by taking out the square root

c2 = 1 + 1

c2 = 2

c = m

If you do this calculation on a calculator, you’ll see that m is approximately equal to 1.4 m.

8

Lesson

3.3.2

The Pythagorean theorem is also useful for finding lengths on graphs that aren’t horizontal or vertical.

Using the Pythagorean TheoremUsing the Pythagorean Theorem

0 1 2 3 4 50

1

2

3

4

y

x

A

B

9

Lesson

3.3.2

Find the length of the line segment KL.

Example 3

Solution follows…

Using the Pythagorean TheoremUsing the Pythagorean Theorem

Solution

Draw a horizontal and vertical line on the plane to make a right triangle.

0 1 2 3 4 50

1

2

3

4

y

x

K

L

2 units

3 units

Solution continues…

Now use the same method as before.

10

Lesson

3.3.2

Find the length of the line segment KL.

Example 3

Using the Pythagorean TheoremUsing the Pythagorean Theorem

Solution (continued)

0 1 2 3 4 50

1

2

3

4

y

x

K

L

2 units

3 unitsKL2 = a2 + b2

KL2 = 32 + 22

Write out the equation

Substitute in the side lengths that you know

Simplify the expression

Cancel out the squaring by taking the square root

KL2 = 9 + 4

KL2 = 13

KL 3.6 units

KL =

11

Lesson

3.3.2

Guided Practice

Using the Pythagorean TheoremUsing the Pythagorean Theorem

Use the Pythagorean theorem to find the length of the hypotenuse in Exercises 1–3.

1. 2. 3.

Solution follows…

c2 = 122 + 52

c2 = 144 + 25c2 = 169 c = 13 ft

12 cm

c ft

5 ft

15 units8 units

c units

c cm3.6 cm

1.5 cm

c2 = 152 + 82

c2 = 225 + 64c2 = 289 c = 17 units

c2 = 3.62 + 1.52

c2 = 12.96 + 2.25c2 = 15.21 c = 3.9 cm

12

Lesson

3.3.2

Guided Practice

Using the Pythagorean TheoremUsing the Pythagorean Theorem

4. Use the Pythagorean theorem to find the length of the line segment XY.

Solution follows…

y

x

–1

–2

0

1

2

3

0 1 2–1–2

X

Y

XY2 = 32 + 32

XY2 = 9 + 9 XY2 = 18 XY2 = XY 4.2 units

13

Lesson

3.3.2

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

a2 = c2 – b2

Subtract b2 from both sidesto get the a2 term by itself.

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.

Using the Pythagorean TheoremUsing the Pythagorean Theorem

14

Lesson

3.3.2

Find the missing leg length in this right triangle.

Example 4

Solution follows…

Using the Pythagorean TheoremUsing the Pythagorean Theorem

a

3 cmSolution

c2 = a2 + b2 First write out the equation

Substitute in the side lengths that you know

Simplify the expression

Take the square root of both sides

a2 = 58 – 9

a2 = 49

a = 7 cm

cm

a2 = c2 – b2 Rearrange it

a =

a2 = – 32

15

Lesson

3.3.2

Guided Practice

Using the Pythagorean TheoremUsing the Pythagorean Theorem

Use the Pythagorean theorem to calculate the missing leg lengths in Exercises 5–8.

5. 6.

7. 8.

Solution follows…

20 cm16 cm

a cm

3.4 ft

1.6 ft a ft

a units

5 units

10 units

a units

units

units

a2 = 202 – 162

a2 = 400 – 256a2 = 144 a = 12 cm

a2 = 3.42 – 1.62

a2 = 11.56 – 2.56a2 = 9 a = 3 ft

a2 = 136 – 102

a2 = 136 – 100a2 = 36 a = 6 units

a2 = 89 – 52

a2 = 89 – 25a2 = 64 a = 8 units

16

Lesson

3.3.2

Independent Practice

Using the Pythagorean TheoremUsing the Pythagorean Theorem

Use the Pythagorean theorem to find the value of c in Exercises 1–5.

1. 2.

4. 5.

Solution follows…

c = 6

12 cm

9 cm

c cm

0.6 m

0.8 m c m

1.5 cm

1 cm

7 in

3.6 m

c m

c = 1c = 15

3.4.8 m

2 in

c in c cmc = c =

17

Lesson

3.3.2

Independent Practice

Using the Pythagorean TheoremUsing the Pythagorean Theorem

Calculate the value of a in Exercises 6–10.

6. 7.

9. 10.

a = 0.9

4 feet

a feet

5 feet

4.5 m

a m 7.5 m

4 cm 4.1 cma = 6

a = 3

8. a cm

3 units

a = 6a units

a in3 in

in

Solution follows…

a =

units

18

Lesson

3.3.2

Independent Practice

Using the Pythagorean TheoremUsing the Pythagorean Theorem

11. Find the length of line AB. 12. Find the perimeter of quadrilateral ABCD

Solution follows…

y

x

A

D 5.1 units

0 1 2 3 4 50

1

2

3

4

y

x

A

B

–1

–2

0

1

2

3

0 1 2–1–2

B

C

2 + 10 12.8 units

19

Lesson

3.3.2 Using the Pythagorean TheoremUsing the Pythagorean Theorem

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.

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