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Solving Similar Figure Applications

Solving 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.

Similar Figures

Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides are in the same ratio.

For example, the two triangles in the figure below are similar. Each side of \(\text{Δ}ABC\) is 4 times the length of the corresponding side of \(\text{Δ}XYZ\).

The above image shows the steps to solve the proportion 1 divided by 12.54 equals 325 divided by p. What are you asked to find? How many Mexican pesos did he get? Assign a variable. Let p equal the number of pesos. Write a sentence that gives the information to find it. If one dollar US is equal to 12.54 pesos, then 325 dollars is how many pesos. Translate into a proportion, be careful of the units. Dollars divided pesos equals dollars divided by pesos to get 1 divided by 12.54 equals 325 divided by p. Multiply both sides by the LCD, 12.54 p to get 1 divided by 12.54 p times 1 divided by 12.54 equals 12.54 p times 325 divided by p. Remove common factors from both sides. Cross out 12.54 from the left side of the equation. Cross out p from the right side of the equation. Simplify to get p equals 4075.5 in the original proportion. Check. Is the answer reasonable? Yes, \$100 would be \$1254 pesos. \$325 is a little more than 3 times this amount, so our answer of 4075.5 pesos makes sense. Substitute p equals 4075.5 in the original proportion. Use a calculator. We now have 1 divided by 12.54 equals 325 divided by p. Next, 1 divided by 12.54 equals 325 divided by 4075.5 to get 0.07874 equals 0.07874. The answer checks.

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.

The above figure shows to similar triangles. The larger triangle labeled A B C. The length of A to B is c, The length of B to C is a. The length of C to A is b. The larger triangle is labeled X Y Z. The length of X to Y is z. The length of Y to Z is x. The length of X to Z is y. To the right of the triangles, it states that measure of corresponding angle A is equal to the measure of corresponding angle X, measure of corresponding angle B is equal to the measure of corresponding angle Y, and measure of corresponding angle C is equal to the measure of corresponding angle Z. Therefore, a divided by x equals b divided by y equals c divided by z.

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

Solve geometry applications.

  1. Read the problem and make 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.

Example

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

The above image shows two similar triangles. Two sides are given for each triangle. The larger triangle is labeled A B C. The length of A to B is 4. The length from B to C is a. The length from C to A is 3.2. The smaller triangle is labeled X Y Z. The length from X to Y is 3. The length from Y to Z is 4.5. The length from Z to X is y.

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 \(\phantom{\rule{0.3em}{0ex}}a=\) 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.\(\phantom{\rule{11.2em}{0ex}}\).\(\phantom{\rule{2.7em}{0ex}}\).
Step 5. Solve the equation.\(\phantom{\rule{11.2em}{0ex}}\).\(\phantom{\rule{2.7em}{0ex}}\).
 \(\phantom{\rule{11.2em}{0ex}}\).\(\phantom{\rule{2.7em}{0ex}}\).
Step 6. Check.
\(\begin{array}{}\begin{array}{ccc}\hfill \frac{4}{3}& \stackrel{?}{=}\hfill & \frac{6}{4.5}\hfill \\ \hfill 4\left(4.5\right)& \stackrel{?}{=}\hfill & 6\left(3\right)\hfill \\ \hfill 18& =\hfill & 18✓\hfill \end{array}\hfill & & & & & \begin{array}{ccc}\hfill \frac{4}{3}& \stackrel{?}{=}\hfill & \frac{3.2}{2.4}\hfill \\ \hfill 4\left(2.4\right)& \stackrel{?}{=}\hfill & 3.2\left(3\right)\hfill \\ \hfill 9.6& =\hfill & 9.6✓\hfill \end{array}\hfill \end{array}\)
 
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.

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

Example

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.

The above image shows two similar triangles and how they are used with maps. The smaller triangle on the left shows San Francisco, Las Vegas and Los Angeles on the three points. San Francisco to Los Angeles is 1.3 inches. Los Angeles to Las Vegas is 1 inch. Las Vegas to San Francisco is 2.1 inches. The second larger triangle shows the same points. The distance from San Francisco to Los Angeles is x. The distance from Los Angeles to Las Vegas is 270 miles. The distance from Las Vegas to San Francisco is not noted.

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.

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

Example

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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