Height of the Flagpole: Using Similar Triangles to Solve Real-World Problems

Have you ever encountered a problem where you need to determine the height of an object, but cannot reach or measure it directly? Such problems are common and can be effectively tackled using the principles of similar triangles. In this article, we will explore a classic example and learn how to use proportions to find the height of a flagpole when provided with the height and shadow length of a classmate.

Understanding Similar Triangles

Similar triangles are triangles that have the same shape and angles but different sizes. The key feature of similar triangles is that their corresponding sides are in proportion. This principle is particularly useful in solving real-world problems involving heights and distances, such as finding the height of a flagpole.

Problem Statement

If your classmate is 6 feet tall and casts a shadow of 4 feet at the same time that the flagpole casts a 20-foot shadow, what is the height of the flagpole?

Solving the Problem Using Proportions

To solve this problem, we can use the concept of similar triangles. The height of the classmate and the length of their shadow create a proportion with the height of the flagpole and its shadow.

Setting Up the Proportion

Let

[ frac{6 : text{feet}}{4 : text{feet}} frac{text{Height of Flagpole}}{20 : text{feet}} ]

By cross-multiplying, we get:

[ 6 times 20 4 times text{Height of Flagpole} ]

Simplifying the equation:

[ 120 4 times text{Height of Flagpole} ]

Dividing both sides by 4:

[ text{Height of Flagpole} frac{120}{4} 30 : text{feet} ]

Therefore, the height of the flagpole is 30 feet.

Formal Problem-Solving Steps

Let’s formalize the problem-solving steps:

Let the height of the flagpole be Set up the proportion: [ frac{6: text{feet}}{4: text{feet}} frac{text{Height of Flagpole}}{20: text{feet}} ] Cross-multiply: [ 6 times 20 4 times text{Height of Flagpole} ] Solve for [ text{Height of Flagpole} frac{6 times 20}{4} 30 : text{feet} ]

The height of the flagpole is 30 feet.

Verification and Application

The solution can be verified by checking that the ratio of the height to the shadow length remains the same. For the classmate, the ratio is (frac{6}{4} 1.5). For the flagpole, the ratio is (frac{30}{20} 1.5), which confirms our solution.

Example Problem with Real-Length Values

Let’s consider another similar problem where the actual height of the classmate is 6 feet and the shadow length is 4 feet. The shadow length of the flagpole is 50 feet. Using the same principle, we can set up the proportional relationship:

[ frac{6}{4} frac{text{Height of Flagpole}}{50} ]

By cross-multiplying, we get:

[ 6 times 50 4 times text{Height of Flagpole} ]

Solving for the height of the flagpole:

[ text{Height of Flagpole} frac{6 times 50}{4} 75 : text{feet} ]

Therefore, the height of the flagpole is 75 feet.

Using the same method, we can solve similar problems by setting up proportions and solving for the unknown height.

Conclusion

Understanding the principle of similar triangles and how to set up and solve proportions is a powerful tool in geometry and real-world problem-solving. Whether you are in class 10 or facing similar challenges, these techniques can help you find the height of tall objects with ease.

Key Takeaways:

Using similar triangles to solve real-world height problems. Setting up and solving proportions to find unknown lengths. Verifying solutions using the ratios of corresponding sides.

Resource Links:

Math is Fun: Similar Triangles Khan Academy: Similarity in Geometry