8.7 Solve proportion and similar figure applications  (Page 4/8)

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

Find the length of side $a$ .

8

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The next example shows how similar triangles are used with maps.

On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. If the actual distance from Los Angeles to Las Vegas is 270 miles find the distance from Los Angeles to San Francisco.

Solution

Read the problem. Draw the figures and label with the given information.	The figures are shown above.
Identify what we are looking for.	The actual distance from Los Angeles to San Francisco.
Name the variables.	Let $x=$ distance from Los Angeles to San Francisco.
Translate into an equation. Since the triangles are similar, the corresponding sides are proportional. We'll make the numerators "miles" and the denominators "inches."
Solve the equation.
Check.
On the map, the distance from Los Angeles to San Francisco is more than the distance from Los Angeles to Las Vegas. Since 351 is more than 270 the answer makes sense.
Answer the question.	The distance from Los Angeles to San Francisco is 351 miles.

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On the map, Seattle, Portland, and Boise form a triangle whose sides are shown in the figure below. If the actual distance from Seattle to Boise is 400 miles, find the distance from Seattle to Portland.

150 miles

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We can use similar figures    to find heights that we cannot directly measure.

Tyler is 6 feet tall. Late one afternoon, his shadow was 8 feet long. At the same time, the shadow of a tree was 24 feet long. Find the height of the tree.

Solution

Read the problem and draw a figure.
We are looking for h , the height of the tree.
We will use similar triangles to write an equation.
The small triangle is similar to the large triangle.
Solve the proportion.
Simplify.
Check.
Tyler's height is less than his shadow's length so it makes sense that the tree's height is less than the length of its shadow.

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Key concepts

• Property of Similar Triangles
  • If $\text{Δ}ABC$ is similar to $\text{Δ}XYZ$ , then their corresponding angle measures are equal and their corresponding sides are in the same ratio.
• Problem Solving Strategy for Geometry Applications
  1. Read the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information.
  2. Identify what we are looking for.
  3. Name what we are looking for by choosing a variable to represent it.
  4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
  5. Solve the equation using good algebra techniques.
  6. Check the answer in the problem and make sure it makes sense.
  7. Answer the question with a complete sentence.