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Using Similar Triangles to Measure Height: A Surveying Example
Using Similar Triangles to Measure Height: A Surveying Example
To solve real-world practical problems in surveying, such as determining the height of a cliff, we can use the principles of similar triangles. This article will guide you through the process, explaining the concept and providing examples.In the given scenario, a surveyor measures the shadows of a cliff and a meter stick to calculate the height of the cliff. This method leverages the properties of similar triangles, which is key in surveying techniques.
Understanding Similar Triangles
Similar triangles have proportional corresponding sides and angles. In surveying, this principle is used to measure heights and distances without directly measuring them. By using a known object (like a meter stick) and comparing its measurements with those of the unknown (e.g., the cliff), we can calculate the required dimensions.
Solving the Height of the Cliff Example
Given Measurements
Let's consider the given problem:
The shadow of the cliff is 12 meters. The shadow of a meter stick is 2 meters.Your goal is to find the height of the cliff.
Setting Up the Proportion
In similar triangles, the corresponding sides are proportional. Let ( h ) be the height of the cliff. Using the given measurements, we set up the following proportion:
[frac{h}{12} frac{1}{2}]Here, the measurement of the cliff's shadow is 12 meters, and the meter stick's shadow is 2 meters, with the height of the meter stick being 1 meter.
Solving for the Height of the Cliff
By cross-multiplying the proportion, we get:
[h cdot 2 1 cdot 12]This simplifies to:
[2h 12]Dividing both sides by 2, we find:
[h frac{12}{2} 6]Therefore, the height of the cliff is 6 meters.
Another Perspective
The problem can also be solved using the concept of proportions with similar triangles directly:
[frac{text{Height of Cliff}}{text{Height of Stick}} frac{text{Shadow of Cliff}}{text{Shadow of Stick}}]Substituting the given values:
[frac{h}{6} frac{1}{2}]Solving for ( h ) again gives the same result:
[h frac{12}{2} 6 , text{meters}]Additional Information
A surveyor, like Mike mentioned, can use the ratio of the known object (the meter stick) to calculate the height of the unknown object (the cliff) based on the principle of similar right-angled triangles. This method is essential in surveying and can be easily applied in other similar scenarios.
Conclusion
The use of similar triangles is a fundamental and practical tool in surveying for measuring distances and heights. By accurately setting up and solving proportions, as demonstrated in the cliff example, surveyors can obtain precise measurements even without direct access to the objects of interest.
Additional Resources
The Math is Fun website provides a comprehensive guide on right-angled triangles, which is essential knowledge for understanding similar triangles and their applications in surveying.