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8.15: Unknown Measures of Similar Figures

Difficulty Level: At Grade Created by: CK-12

Let's Think About It

License: CC BY-NC 3.0


Shelby's father is a former Olympic swimmer. He loves swimming, but he cannot build an Olympic sized swimming pool in his backyard because he does not have the space. He decides to build a smaller pool that is similar to the dimensions of an Olympic pool. An Olympic pool has a length of 50 meters and a width of 25 meters. His pool has a length of 25 meters. What is the width of his pool?

In this concept, you will learn how to find the unknown measures of similar figures.

Guidance

If you know the length of a side in one figure, you can use the scale factor to find the measure of the corresponding side in a similar figure. Let’s see how this works.

Side a in triangle ABC corresponds to side x in the smaller triangle XYZ. Side x is 4 meters long and the scale factor is 6. What is the measure of side a?

You have been told that two sides, a and x, correspond in a small triangle and a large one. Side x is 4 meters long, and the scale factor tells you that side a will be six times longer. Let’s write this out and solve.

side x×scale factor4×624 m=side a=side a=side a

Side a must have a length of 24 meters.

You can check by setting up a ratio that compares the lengths of the two sides. If the scale factor is 6, then your work is accurate.

ax=244=6

Sometimes, you can figure out missing side lengths by looking at the given measures. Always look at the diagram of the figures and see if you can determine the missing length without measuring.

Look at these two rectangles. First, look and see if you can figure out the relationship between the two figures. To do this, you compare the side lengths of each part of the two figures.

You need to figure out the measurement of side GH in the second rectangle.

You can see that the measurements in the second rectangle are half as big as the measurements in the first. Also, you know that the opposite sides of a rectangle are congruent. Therefore, the missing side length is 4.

Similar figures that are related by a scale factor are often seen in maps, architectural blueprints and diagrams. In most of these cases, the scale factor is given so that you know how to enlarge the items in the drawing to their real sizes. Take a look at the floor plan below. It shows where the furniture is located in a living room.

It tells you that one inch in the drawing is equal to two feet in actual size. Therefore, if you know the size in inches of any object in the floor plan, you can find its actual size in feet. Let’s give it a try.

Find the sofa on the floor plan then find its length in inches. The sofa in the floor plan is 2 inches long. This is like knowing the length of one side in a similar figure. Now use the scale factor as you would to find the length of the corresponding side in a similar figure (in this case the “corresponding side” is the actual sofa). Simply multiply the length you know by the scale factor:

sofa drawing×scale factor2 inches×2=actual sofa size=4 feet

The sofa is 4 feet long.

Next, let's calculate the real length of the fireplace.Use a ruler to measure the fireplace in the drawing. It is 2.5 inches long. Multiply this by the scale factor to find the length in feet.

fireplace drawing×scale factor2.5 inches×2=actual fireplace length=5 feet

The real length of the fireplace is 5 feet.

You can also reverse the process to take an actual size and reduce it.

Guided Practice

Use the scale factor of the similar figures below to find the measure of KJ.

First, set up a proportion to solve for the missing side.

KJ5=64

This proportion is written so that the corresponding sides form the two ratios of the proportion. You can say that KJ is the unknown in this proportion.

Next, cross multiply. 

KJ×45×64KJ=4KJ=30=30

Then solve the equation for KJ by dividing both sides of the equation by 4.

30÷4KJ=7.5=7.5

The answer is that the side length of KJ is 7.5.

Examples

Example 1

Chris is making a drawing of his school and the grounds around it. The basketball court is 75 feet long and 40 feet wide. If Chris uses a scale factor in which 1 inch equals 10 feet, what should the dimensions of the basketball court be in his drawing?

First, write the information that you know.

The actual size of the basketball court and the scale factor Chris is using for his drawing.

Next, set up an equation to find the length Chris should draw. 

drawing length×scale factordrawing length×10drawing lengthdrawing length=actual basketball court length=75 feet=75÷10=7.5 inches

The length of the basketball court in Chris’s drawing should be 7.5 inches.

Then, use the same process to find the width Chris should draw.

drawing width×scale factordrawing width×10drawing widthdrawing width=actual basketball court width=40 feet=40÷10=4 inches

The answer is that Chris should represent the basketball court as a 7.5 by 4 inch rectangle on his drawing.

Example 2

Solve for x.

34=x12

First, cross multiply.

4x = 36

Next, divide both sides of the equation by 4.

x = 9

Then, state the solution.

9

The answer is that x equals 9.

Example 3

Solve for x.

36=1x

First, cross multiply.

3x = 6

Next, divide both sides of the equation by 3.

x = 2

Then, state the solution.

2

The answer is that x equals 2.

Follow Up

License: CC BY-NC 3.0

Remember Shelby's father and his swimming pool? He builds a pool that is similar to an Olympic pool. An Olympic pool has a length of 50 meters and a width of 25 meters. If his pool has a length of 25 meters, what is the width of his pool?

First, set up an equation that can be used to solve for the width of his pool.

 5025=25x

Next, cross multiply.

 50x=625

Then, divide both sides of the equation by 50.

 x= 12.5

The answer is that his pool has a width of 12.5 meters.

Video Review

Explore More

Solve each problem.

1. Side m in triangle LMN corresponds to side c in the smaller triangle BCD. Side m is 12 cm long and the scale factor is 4. What is the measure of side c?

2. Side q in triangle PQR corresponds to side y in the smaller triangle XYZ. Side y is 8 inches long and the scale factor is 7. What is the measure of side q?

Solve each proportion for the missing side length.

3. 710=x30

4. 1.53=x6

Now use the scale factor to create a new ratio.

5. 13, scale factor 4

6. 85, scale factor 5

7. 93, scale factor 3

8. Find the scale factor of the similar figures below and then use it to find the measure of LO.

9. Use the scale factor of the similar figures below to find the measure of JK.

Use the map below and a ruler to answer the questions that follow.

10. How far does Delia live from her school?

11. How far is it from the library to the park?

12. How far does Delia live from City Hall?

13. Delia drew another point to show the police station on her map. She drew it 1.5 inches away from the City Hall. What is the actual distance between the police station and City Hall?

14. How far does Delia live from the park?

15. How far does she live from the library?

16. What is the farthest that Delia will travel to any one item on her map?

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 8.15. 

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0

Description

Difficulty Level:

At Grade

Grades:

Date Created:

Dec 02, 2015

Last Modified:

Dec 02, 2015
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