Understanding Similar Triangles: A Geometry Application in Solving Heights and Shadows

When faced with real-world problems such as determining the height of a building based on the shadows they cast, the concept of similar triangles becomes a powerful tool. This article will explore how to use similar triangles to find the solution to the problem at hand: a tree that is 12 feet tall casts a 9-foot shadow, and a nearby building casts a 21-foot shadow. By applying geometric principles, we will determine the height of the building.

Introduction to Similar Triangles

Similar triangles are two triangles that have the same shape but not necessarily the same size. This means that the corresponding angles are equal, and the corresponding sides are proportional. This property allows us to set up proportional relationships that help us find unknown measurements.

Solving the Problem Using Similar Triangles

Given the information:

- Tree height: 12 feet

- Tree shadow length: 9 feet

- Building shadow length: 21 feet

We need to find the height of the building. Here's the step-by-step process:

Step 1: Setting Up the Proportion

According to the properties of similar triangles, the ratio of the height of the tree to the length of its shadow is equal to the ratio of the height of the building to the length of its shadow. This can be represented as:

$frac{12 text{ ft}}{9 text{ ft}} frac{h}{21 text{ ft}}$

Step 2: Cross-Multiplication

To solve for the height of the building, we cross-multiply:

$12 text{ ft} times 21 text{ ft} 9 text{ ft} times h$

$252 text{ ft}^2 9h$

Step 3: Solving for (h)

Divide both sides of the equation by 9:

$h frac{252 text{ ft}^2}{9 text{ ft}}$

$h 28 text{ ft}$

Therefore, the height of the building is (28 text{ ft}), making option C the correct answer.

Alternative Methods for Solving the Problem

There are several methods to solve this problem, and each can offer insights into different mathematical principles and their applications.

Method 2: Ratio Analysis

The tree's shadow is (frac{12}{9} frac{4}{3}) times the height of the tree. Similarly, the building's shadow is (7 text{ times}) (since (frac{21}{9} frac{7}{3} 7/3)) the size of the tree's shadow. Therefore, the building's height will also be (7/3) times the tree's height:

$frac{4}{3} times 21 28 text{ ft}$

Method 3: Trigonometry

We can use trigonometric ratios, specifically the tangent function, to solve for the angle of elevation and then use it to find the height of the building. The angle of elevation can be calculated using the tangent function:

$tan(theta) frac{12}{9} frac{4}{3}$

Using a table of tangents or trigonometric values, we find that the angle (theta) is approximately 53 degrees. Using this angle, we can apply it to the building's shadow:

$tan(53) frac{text{Height of building}}{21}$

$frac{4}{3} frac{h}{21}$

$h 28 text{ ft}$

Conclusion

The height of the building is (28 text{ ft}), confirming that option C is the correct answer. By understanding and applying the principles of similar triangles, trigonometry, and proportional reasoning, we can confidently solve real-world geometric problems.

Keywords: similar triangles, shadow, geometry, trigonometry