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8.6: Indirect Measurement

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Introduction

The Angel Sculpture

In the field behind the museum, Mrs. Gilson showed the students a replica of a sculpture known as the “Angel of the North.” The students went outside to have a better look at the huge structure.

“It is huge,” Carmen said to Henry.

“Yup, I wonder how tall it is?” he whispered.

“You can figure that out quite easily with math,” Mrs. Gilson said, overhearing the conversation.

“How can I do that?” Henry asked.

“How tall are you?”

“Five feet,” Henry answered.

“Good, and it looks like your shadow is about half as long as your height. Can you figure it out now?”

Henry looked at Carmen and then back at Mrs. Gilson.

“I have an idea,” he said smiling.

Do you have an idea what Henry is thinking about? If you pay attention to this lesson, you will know how to figure out how tall the statue is. We will come back to Henry, Carmen and the sculpture at the end of the lesson.

What You Will Learn

In this lesson you will learn the following skills:

  • Recognize the ratio of corresponding side lengths of similar figures as the scale factor.
  • Use proportions to find unknown measures of corresponding parts of similar figures.
  • Use diagrams of similar figures to find unknown lengths without directly measuring.
  • Solve real-world problems involving indirect measurement.

Teaching Time

I. Recognize the Ratio of Corresponding Side Lengths of Similar Figures as the Scale Factor

Similar figures are shapes that exist in proportion to each other. They have congruent angles, but their sides are different lengths. Squares, for example, are similar to each other because they always have four 90^{\circ} angles and four equal sides, even if the lengths of their sides differ. Other figures can be similar too, if their angles are equal. Let’s look at some pairs of similar figures.

Notice that in each pair the figures look the same, but one is smaller than the other. As you can see, similar figures have congruent angles but sides of different lengths.

Even though similar figures have sides of different lengths, corresponding sides still have a relationship with each other. Each pair of corresponding sides has the same relationship as every other pair of corresponding sides, so that, altogether, the pairs of sides exist in proportion to each other. For instance, if a side in one figure is twice as long as its corresponding side in a similar figure, all of the other sides will be twice as long too.

In this lesson, we are going to use these relationships to find the measures of unknown sides. This method is called indirect measurement.

Let’s take a step back to similar figures to understand how indirect measurement works.

First, let’s make sure we can recognize corresponding parts of similar figures. Similar figures have exactly the same angles. Therefore each angle in one figure corresponds to an angle in the other.

These triangles are similar because their angles have the same measures. Which corresponds to which? Angle B is 100^{\circ}. Its corresponding angle will also measure 100^{\circ}: that makes angle Q its corresponding angle. Angles A and P correspond, and angles C and R correspond.

Similar figures also have corresponding sides, even though the sides are not congruent. Corresponding sides are not always easy to spot. We can think of corresponding sides as those which are in the same place in relation to corresponding angles. For instance, side AB, between angles A and B, must correspond to side PQ, because A corresponds to P and B corresponds to Q.

Corresponding sides also have lengths that are related, even though they are not congruent. Specifically, the side lengths are proportional. In other words, each pair of corresponding sides has the same ratio as every other pair of corresponding sides. Take a look at the rectangles below.

These rectangles are similar because the sides of one are proportional to the other. We can see this if we set up proportions for each pair of corresponding sides. Let’s put the sides of the large rectangle on the top and the corresponding sides of the small rectangle on the bottom.

\frac{LM}{WX} &= \frac{8}{4}\\\frac{MN}{XY} &= \frac{6}{3}\\\frac{ON}{ZY} &= \frac{8}{4}\\\frac{LO}{WZ} &= \frac{6}{3}

Now you can clearly see each relationship. To figure out if the pairs do indeed form a proportion, we have to divide the numerator by the denominator. If the quotient is the same, then the ratios each form the same proportion and the figures are similar.

\frac{LM}{WX} &= \frac{8}{4} = 2\\\frac{MN}{XY} &= \frac{6}{3} = 2\\\frac{ON}{ZY} &= \frac{8}{4} = 2\\\frac{LO}{WZ} &= \frac{6}{3} = 2

Each quotient is the same so these ratios are proportional. The side lengths are proportional and the figures are similar.

We call this the scale factor. The scale factor is the ratio that determines the proportional relationship between the sides of similar figures. For the pairs of sides to be proportional to each other, they must have the same scale factor. In other words, similar figures have congruent angles and sides with the same scale factor. A scale factor of 2 means that each side of the larger figure is twice as long as its corresponding side is in the smaller figure.

Example

What is the scale factor of the figures below?

We need to find the scale factor, so let’s set up the proportions of the sides. Let’s put all the sides from the large figure on top and the sides from the small figure on the bottom. It doesn’t matter which we put on top, as long as we keep all the sides from one figure in the same place.

\frac{QR}{HI} &= \frac{15}{5}\\\frac{TS}{KJ} &= \frac{21}{7}\\\frac{RS}{IJ} &= \frac{6}{3}\\\frac{QT}{HK} &= \frac{15}{5}

Now all we have to do is divide to find the scale factor.

\frac{QR}{HI} &= \frac{15}{5} = 3\\\frac{TS}{KJ} &= \frac{21}{7} = 3\\\frac{RS}{IJ} &= \frac{6}{3} = 3\\\frac{QT}{HK} &= \frac{15}{5} = 3

The scale factor for these similar figures is 3. This means that the sides of the first quadrilateral are exactly three times bigger than the second.

What happens if we had written the numbers the other way around?

If we had put the measurements of the smaller figure on top and the larger figure on the bottom, we would have found a scale factor of \frac{1}{3}. You can write them the other way too as long as you understand how to read the scale factor. This is another way of saying the scale factor is three. When we say that the larger figure is three times as big as the small one, it’s the same as saying that the small figure is one-third the size of the larger one.

8L. Lesson Exercises

Look at each ratios and determine the scale factor.

  1. \frac{18}{6} and \frac{24}{8}
  2. \frac{12}{6} and \frac{8}{4}
  3. \frac{25}{5}, \frac{45}{9}, \frac{15}{3}

Take a few minutes to check your answers with a peer.

II. Use Proportions to Find Unknown Measures of Corresponding Parts of Similar Figures

We can use a scale factor to help us to determine unknown measures. We don’t use the scale factor alone we apply it to the proportion. If we know the length of a side in one figure, we can use the scale factor to find the measure of the corresponding side in a similar figure. Let’s see how this works.

Example

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?

We have been told that two sides, a and x, correspond in a small triangle and a large one. If we know the length of one and the scale factor, we can find the length of the other. Side x is 4 meters long, and the scale factor tells us that side a will be six times as long. Let’s write this out and solve.

\text{side} \ x  \times  \text{scale factor}  &=  \text{side} \ a\\4  \times  6  &=  \text{side} \ a\\24 \ m  &=  \text{side} \ a

Side a must have a length of 24 meters.

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

\frac{a}{x} = \frac{24}{4} = 6

Now we know that our work is accurate.

Example

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

Now the first thing that we can do is to set up a proportion to solve for the missing side. Remember that a proportion is two equal ratios. We can set up and compare the corresponding sides.

Here is our proportion.

\frac{KJ}{5} = \frac{6}{4}

Our proportion is written so that the corresponding sides form the two ratios of the proportion. We can say that KJ is our unknown in this proportion.

Do you remember how to solve proportions?

We can see a clear relationship between five and four, so we need to use cross products.

KJ \times 4 &= 4KJ\\5 \times 6 &= 30\\4KJ &= 30

Now we can solve the equation for KJ by dividing both sides of the equation by 4.

30 \div 4 &= 7.5\\KJ &= 7.5

The side length of KJ is 7.5.

III. Use Diagrams of Similar Figures to Find Unknown Lengths without Directly Measuring

Sometimes, you can figure out missing side lengths just by looking at the given measures. In the last example it was too tricky because we couldn’t see the relationship between 6 and 4. However this isn’t always the case. Always look at the diagram of the figures and see if you can determine the missing length without measuring.

Let’s look at an example where this is the case.

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

We 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.

Working in this way can often save you some time!

IV. Solve Real-World Problems Involving Indirect Measurement

You may be surprised how often we use similar figures that are related by a scale factor. Maps, architectural blueprints, and diagrams are just some examples. In most of these cases, the scale factor is given so that we 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.

The size of everything in the drawing has been made smaller from a real size by the scale factor. What is the scale factor for the floor plan?

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

How many feet long is the sofa?

Let’s find the sofa on the floor plan. Then we can use a ruler to find its length in inches. How many inches long is the drawing of the sofa? The sofa in the floor plan is 2 inches long. Imagine this is like knowing the length of one side in a similar figure. Now we need to use the scale factor as we would to find the length of the corresponding side in a similar figure (in this case the “corresponding side” is the actual sofa). We simply multiply the length we know by the scale factor:

\text{sofa drawing} \times \text{scale factor}  &=  \text{actual sofa size}\\2 \ inches  \times  2  &=  4 \ feet

The sofa is four feet long.

How long is the fireplace?

Use a ruler to measure the fireplace in the drawing. It is 2.5 inches long. We multiply this by the scale factor to find the length in feet.

\text{fireplace drawing} \times \text{scale factor} &= \text{actual fireplace length}\\2.5 \ inches \times 2 &= 5 \ feet

The real length of the fireplace is 5 feet.

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

Example

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 of all, what do we need to find?

We need to know the dimensions (length and width) that the small version of the basketball court should be.

What information have we been given?

We know the actual size of the basketball court, and we know the scale factor Chris is using for his drawing. We can set up an equation to find the drawn dimensions. We’ll have to find the length first and then the width.

\text{drawing length} \times \text{scale factor}  &=  \text{actual basketball court length}\\\text{drawing length} \times 10  &=  75 \ feet\\	\text{drawing length}  &=  75  \div  10\\\text{drawing length}  &=  7.5 \ inches

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

Now let’s use the same process to find the width Chris should draw.

\text{drawing width} \times \text{scale factor}  &=  \text{actual basketball court width}\\\text{drawing width}  \times  10  &=  40 \ feet\\	\text{drawing width} &= 40  \div  10\\\text{drawing width}  &=  4 \ inches

Great! Now we know that Chris should represent the basketball court as a 4 by 7.5 inch rectangle on his drawing.

Real-Life Example Completed

The Angel Sculpture

Here is the original problem once again. Reread it and underline any important information.

In the field behind the museum, Mrs. Gilson showed the students a replica of a sculpture known as the “Angel of the North.” The students went outside to have a better look at the huge structure.

“It is huge,” Carmen said to Henry.

“Yup, I wonder how tall it is?” he whispered.

“You can figure that out quite easily with math,” Mrs. Gilson, said overhearing the conversation.

“How can I do that?” Henry asked.

“How tall are you?”

“Five feet,” Henry answered.

“Good, and it looks like your shadow is about half as long as your height. Can you figure it out now?”

Henry looked at Carmen and then back at Mrs. Gilson.

“I have an idea,” he said, smiling.

Let’s think about how Henry and Carmen could figure out the height of the statue. We know that Henry is five feet tall and that his shadow is half as long as he is tall. Now we can write a ratio to compare Henry’s height to his shadow’s length.

\frac{Henry' s \ height}{Shadow' s \ length} = \frac{5 \ feet}{2.5 \ feet}

Next, we figure out the height of the statue. Henry and Carmen figure out very quickly that they need to learn the length of the shadow of the statue to figure out the height of the statue. Once they know the length of the shadow, they can use proportional reasoning and indirect measurement to figure out the statue’s height.

Approximating 1 foot using a length a little longer than Henry’s sneaker, they measure 32 \frac{1}{2} feet. It is not an exact measure, but they feel that it is very close.

Now they write the following proportion.

\frac{5 \ ft}{2.5 \ ft} = \frac{x}{32.5 \ ft}

Taking out a notebook, Carmen cross multiplies to solve the proportion.

5(32.5) &= 2.5x\\162.5 &= 2.5x\\x &= 65

The sculpture is approximately 65 feet tall.

After completing their work, Henry and Carmen check out their answer with the curator of the museum. The statue is actually 65.6 feet tall. Their work was very close to accurate! Indirect measurement was very useful!

Vocabulary

Here are the vocabulary words that are found in this lesson.

Similar Figures
Figures that have the same angle measures but not the same side lengths.
Indirect Measurement
using relationships between side lengths to figure out missing measures.
Scale Factor
the proportional relationship between two side lengths.
Proportions
two equal ratios

Technology Integration

Khan Academy Scale and Indirect Measurement

James Sousa, Indirect Measurement Using Similar Triangles

James Sousa, Solving for Unknown Values Using the Properties of Similar Triangles

Other Videos:

  1. http://www.onlinemathlearning.com/indirect-measurement.html – Here is a video on indirect measurement.

Time to Practice

Directions: Find the scale factor of the pairs of similar figures below.

Directions: Use each ratio to determine scale factor.

5. \frac{3}{1}

6. \frac{8}{2}

7. \frac{2}{8}

8. \frac{10}{5}

9. \frac{12}{4}

10. \frac{16}{2}

Directions: Solve each problem.

11. 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?

12. 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?

Directions: Solve each proportion for the missing side length.

13. \frac{3}{4} = \frac{x}{12}

14. \frac{3}{6} = \frac{1}{x}

15. \frac{5}{8} = \frac{1}{x}

16. \frac{7}{10} = \frac{x}{30}

17. \frac{1.5}{3} = \frac{x}{6}

Directions: Now use the scale factor to create a new ratio.

18. \frac{1}{3}, scale factor 4

19. \frac{8}{5}, scale factor 5

20. \frac{9}{3}, scale factor 3

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

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

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

22. How far does Delia live from her school?

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

24. How far does Delia live from City Hall?

25. 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?

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