## Solving Similar Figure Applications

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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 \(\text{Δ}XYZ\).

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.

### Solve geometry applications.

**Read**the problem and make all the words and ideas are understood. Draw the figure and label it with the given information.**Identify**what we are looking for.**Name**what we are looking for by choosing a variable to represent it.**Translate**into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.**Solve the equation**using good algebra techniques.**Check**the answer in the problem and make sure it makes sense.**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.

### 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.

### 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. | |