8.7 Solve proportion and similar figure applications  (Page 3/8)

In the example above, we related the number of pesos to the number of dollars by using a proportion. We could say the number of pesos is proportional to the number of dollars. If two quantities are related by a proportion, we say that they are proportional.

Solve similar figure applications

When you shrink or enlarge a photo on a phone or tablet, figure out a distance on a map, or use a pattern to build a bookcase or sew a dress, you are working with similar figures    . If two figures have exactly the same shape, but different sizes, they are said to be similar. One is a scale model of the other. All their corresponding angles have the same measures and their corresponding sides are in the same ratio.

For example, the two triangles in [link] are similar. Each side of $\text{Δ}ABC$ is 4 times the length of the corresponding side of $\text{Δ}XYZ$ .

This is summed up in the Property of Similar Triangles.

Property of similar triangles

If $\text{Δ}ABC$ is similar to $\text{Δ}XYZ$ , then their corresponding angle measure are equal and their corresponding sides are in the same ratio.

To solve applications with similar figures we will follow the Problem-Solving Strategy for Geometry Applications we used earlier.

$\text{Δ}ABC$ is similar to $\text{Δ}XYZ$ . The lengths of two sides of each triangle are given. Find the lengths of the third sides.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.	Figure is given.
Step 2. Identify what we are looking for.	the length of the sides of similar triangles
Step 3. Name the variables.	Let $a=$ length of the third side of $\text{Δ}ABC.$    $y=$ length of the third side of $\text{Δ}XYZ$
Step 4. Translate.	Since the triangles are similar, the corresponding sides are proportional.
We need to write an equation that compares the side we are looking for to a known ratio. Since the side AB = 4 corresponds to the side XY = 3 we know $\frac{AB}{XY}=\frac{4}{3}$ . So we write equations with $\frac{AB}{XY}$ to find the sides we are looking for. Be careful to match up corresponding sides correctly.	$\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}.$
Substitute.	$$ $$
Step 5. Solve the equation.	$$ $$
	$$ $$
Step 6. Check. $\begin{array}{}\\ \\ \begin{array}{ccc}\hfill \frac{4}{3}& \stackrel{?}{=}\hfill & \frac{6}{4.5}\hfill \\ \hfill 4(4.5)& \stackrel{?}{=}\hfill & 6(3)\hfill \\ \hfill 18& =\hfill & 18✓\hfill \end{array}\hfill & & & & & \begin{array}{ccc}\hfill \frac{4}{3}& \stackrel{?}{=}\hfill & \frac{3.2}{2.4}\hfill \\ \hfill 4(2.4)& \stackrel{?}{=}\hfill & 3.2(3)\hfill \\ \hfill 9.6& =\hfill & 9.6✓\hfill \end{array}\hfill \end{array}$
Step 7. Answer the question.	The third side of $\text{Δ}ABC$ is 6 and the third side of $\text{Δ}XYZ$ is 2.4.

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