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Indirect Measurement

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Recognizing Similarity
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Have you ever thought about shadows? Take a look at this dilemma.

A person is five feet tall and casts a shadow of 2 feet. A tower casts a shadow that is 10 feet long. What is the height of the tower?

Do you know how to figure this out?

Pay attention and you will learn how to accomplish this task in this Concept.

Guidance

Congruent means exactly the same, having the same size and shape.

Sometimes, a figure will have the same shape, but not the same size. It will be either smaller or larger than the original figure. When this happens, we say that the two figures are 'similar.

Similar figures have the same shape, but not the same size.

Think about this for a minute, if a figure has the same shape, but not the same size, then there is still a relationship between the two figures. The relationship is created based on the shape being the same.

That is a good question.

Let’s start by thinking about angles. With similar figures, each angle of one figure in a similar pair corresponds and is congruent to an angle in the other.

For instance, the top point of one triangle corresponds to the top point of the other triangle in a similar pair. We call these corresponding parts.

Notice that the angles match in these two triangles. The shape of the triangles is the same, and you can see that the angles do match.

What about the side lengths?

The sides in similar pairs also correspond to each other (such as the base of each triangle), but they are not congruent; they are proportional. We can determine whether figures are similar to each other by comparing their corresponding parts. Corresponding parts are especially helpful when one figure is rotated so that it is not clear which angles and sides correspond to which in the other figure.

Now let’s look at the corresponding side lengths. In the first rectangle, the short side is 4 and the long side is 8. We know that opposite sides of a rectangle are congruent, so we don’t need to worry about writing measurements on the other two sides. We can compare the measurements in the first rectangle with the ones in the second rectangle. In the second rectangle, the short side is 2 and the long side is 4.

Let’s write a proportion to compare the corresponding sides.

\frac{short \ side}{short \ side} &= \frac{long \ side}{long \  side}\\\frac{4}{2} &= \frac{8}{4}

You can see that these two ratios form a proportion. You can use this information to prove whether or not two figures are similar as well. Remember, the angle measures must be the same, and the side lengths must be proportional.

Write these notes on similar figures down in your notebooks.

Now that you understand how to identify whether or not two figures are similar, we can look similar triangles. Similar triangles are very useful because we can use them to figure out measurements. Many years ago, this is how people used to figure out the measurements for things that were too high or big to measure. They used indirect measurement. Indirect measurement uses similar triangles and proportions to figure out lengths or distances.

But first, let’s think about similar triangles.

Similar triangles have the same properties as other similar figures. The angle measures are the same and the corresponding side lengths are proportional. Let’s look at this diagram to understand this.

Now we can compare the angles and corresponding side lengths. Let’s begin with the angles.

\angle A & \cong \angle D\\\angle B & \cong \angle E\\\angle C & \cong \angle F

Next, we can look at the corresponding side lengths. In the diagram, we haven’t been given any measurements, but we can use the lowercase letters to show which sides correspond.

\frac{a}{d}=\frac{b}{e}=\frac{c}{f}

This shows that the side lengths form a ratio and that each of these is proportional to the other.

We can use this information when problem solving for missing side lengths.

That is a good question. First, we would have to know some of the side lengths. Let’s assign some lengths to the sides in the diagram above.

a &= 12\\b &= ?\\c &= 3\\d &= 4\\e &= 3\\f &= 1

Now we can take these given measures and substitute them into the proportion that we wrote earlier. Notice that we don’t have the measure of side b , so we will need to solve for that missing measurement.

\frac{12}{4}=\frac{b}{3}=\frac{3}{1}

Next, we can use two of the three ratios to solve the proportions. We have three ratios, but we don’t need all three because two equal ratios form a proportion. This means that we only need to work with two ratios to solve for the value of b .

\frac{12}{4}=\frac{b}{3}

Now we can cross - multiply and solve the proportion.

4b &= 36\\b &= 9

The value of b is 9.

The key to working with indirect measurement is to always be clear about what is being compared. You write your ratios and then form a proportion and solve for the missing length or distance.

Write some notes about indirect measurement down in your notebook.

Solve each for the missing value.

Example A

\frac{2}{3}=\frac{4}{6}=\frac{x}{18}

Solution:  x = 12

Example B

\frac{4}{5}=\frac{12}{15}=\frac{x}{30}

Solution:  x = 24

Example C

\frac{8}{9}=\frac{16}{x}=\frac{32}{36}

Solution:  x = 18

Now let's go back to the dilemma from the beginning of the Concept.

This may seem like a very challenging problem to solve, however if you think about people and shadows as they are related to triangles it becomes much easier. Look at this picture.

Notice that the person and the shadow form two sides of a triangle and we can draw an imaginary line from the head of the person to the tip of the shadow. Shadows are a way of working with triangles and indirect measurement. In fact, you will often hear these types of problems referred to as shadow problems.

To solve this one, let’s figure out how to use similar triangles to figure out the height of the tower. First, think about what is being compared. We are comparing the height of the person with the length of the shadow. That is the first ratio.

\frac{person}{shadow}=\frac{5 \ ft}{2 \ ft}

Next, we look at the tower. We don’t know the height of the tower that is our variable. We do know the length of the shadow. Here is our second ratio.

\frac{tower}{shadow}=\frac{x}{10 \ ft}

We can say that these two triangles are similar and that similar triangles are proportional. Therefore, these two ratios form a proportion. Let’s write them as a proportion.

\frac{5 \ ft}{2 \ ft}=\frac{x}{10 \ ft}

Now we can cross - multiply and solve the proportion.

2x &= 50 \ ft\\x &= 25 \ ft

The tower is 25 feet tall.

Vocabulary

Congruent
having the same size, shape and measure.
Similar
having the same shape, but not the same size. Angle measures are the same and side lengths are proportional.
Proportional
the side lengths create ratios that form a proportion.
Indirect Measurement
using similar triangles to figure out challenging distances or lengths.

Guided Practice

Here is one for you to try on your own.

\frac{1}{4}=\frac{2}{8}=\frac{24}{x}

Solution

First, we can look at the relationship between the numerators.

One was multiplied by two to get two. Then two was multiplied by 12 to get 24.

Now let's look at the denominators.

Four was multiplied by two to get 8. Then we need to multiply 8 by 12 to find the missing denominator.

Our answer is x = 96 .

Practice

Directions: Identify whether or not each pair of triangles is similar based on the ratios of their sides.

  1. Triangle A has side lengths of 2, 4, and 6. Triangle B has side lengths of 6, 12 and 24. Are these triangles similar?
  2. Triangle C has side lengths of 4, 5, and 10. Triangle B has side lengths of 2, 2.5 and 5. Are these two triangles similar?
  3. Triangle D has side lengths of 5, 8, and 12. Triangle B has side lengths of 10, 16 and 24. Are these two triangles similar?
  4. Triangle A has side lengths of 10, 12, and 14. Triangle B has side lengths of 5, 7 and 9. Are these two triangles similar?
  5. Triangle B has side lengths of 8, 14, and 20. Triangle C has side lengths of 4, 7 and 10. Are these two triangles similar?
  6. Triangle E has side lengths of 20, 11 and 8. Triangle F has side lengths of 10, 5.5 and 5. Are these two triangles similar?
  7. Triangle G has side lengths of 6, 8 and 12. Triangle H has side lengths of 18, 24 and 36. Are these two triangles similar?
  8. Triangle I has side lengths of 8, 12, and 16. Triangle J has side lengths of 4, 8 and 10. Are these two triangles similar?

Directions: Find the missing length by looking at each series of ratios. The top value represents the side lengths of the first similar triangle. The bottom value represents the side lengths of the second similar triangle.

  1. \frac{1}{2}=\frac{3}{6}=\frac{9}{x}
  2. \frac{3}{6}=\frac{6}{12}=\frac{10}{x}
  3. \frac{4}{2}=\frac{10}{x}=\frac{12}{6}
  4. \frac{6}{2}=\frac{9}{x}=\frac{12}{4}
  5. \frac{5}{10}=\frac{10}{20}=\frac{15}{x}
  6. \frac{12}{6}=\frac{20}{10}=\frac{15}{x}
  7. \frac{16}{x}=\frac{20}{5}=\frac{24}{6}

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