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Finding Actual Distance on a Map Using Similar Triangles
Apply the property of similar triangles to convert a measurement on a map into a real-world distance.
Problem: On a map, San Francisco, Las Vegas, and Los Angeles form a triangle. The map distances are: Los Angeles to San Francisco inches, Los Angeles to Las Vegas inch, and Las Vegas to San Francisco inches. If the actual distance from Los Angeles to Las Vegas is miles, find the distance from Los Angeles to San Francisco.
- Read the problem and draw the figures. The small triangle represents the map distances (in inches) and the large triangle represents the actual distances (in miles).
- Identify: The actual distance from Los Angeles to San Francisco.
- Name: Let the distance from Los Angeles to San Francisco.
- Translate: Because the map triangle and the real-world triangle are similar, their corresponding sides are proportional. Place miles in the numerators and inches in the denominators:
- Solve: Multiply both sides by :
- Check: On the map, the Los Angeles–San Francisco side ( inches) is longer than the Los Angeles–Las Vegas side ( inch), so the actual distance should also be larger: ✓. Substituting back: and ✓.
- Answer: The distance from Los Angeles to San Francisco is miles.
This example demonstrates how a map acts as a scaled-down similar figure: each inch on the map corresponds to the same number of miles in reality. Setting up the proportion with consistent units — miles over inches on both sides — ensures the correct pairing of corresponding sides.
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Learn After
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