Problem 31 - Introductory Algebra for College Students

Question

Use similar triangles to solve. A tree casts a shadow 12 feet long. At the same time, a vertical rod 8 feet high casts a shadow 6 feet long. How tall is the tree? (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

The tree is 16 feet tall.

Step 1: Identify the corresponding sides in the two triangles

In the first triangle, the height of the tree is the side opposite to the right-angle and the length of the shadow is the side adjacent to the right-angle. Similarly, in the second triangle, the height of the rod is the side opposite to the right-angle and the length of the shadow is the side adjacent to the right-angle. These sides are corresponding in the two triangles.

Step 2: Set up the proportion

For similar triangles, the ratio of corresponding sides is equal. Hence, the ratio of the height of the tree (T) to the length of its shadow (12 feet) is equal to the ratio of the height of the rod (8 feet) to the length of its shadow (6 feet). This gives: \(\frac{T}{12} = \frac{8}{6}\)

Step 3: Solve the proportion for T

Cross-multiplying the equation will give us: \(T =\frac{8}{6} \times 12\). Now multiplying the ratios gives: \(T = 16\) feet.

Key Concepts

Proportions: Understanding Ratios in Geometry

Proportions in mathematics are a way to express that two ratios or fractions are equivalent. When we talk about similar triangles, proportions become a handy tool that ties everything together. Similar triangles, by definition, have the same shape but different sizes. This means all their corresponding angles are equal, and their sides are in proportion to each other.
To set up a proportion, we compare two ratios of corresponding sides. In our original exercise, the height of the tree is to its shadow as the height of the rod is to its shadow. We can write this relationship as:

• Tree's height / Tree's shadow = Rod's height / Rod's shadow

This equation shows that the heights relate to the shadows in the same way, which helps us find the missing length. This approach works since the triangles formed by the tree or rod with their shadows are similar. So, understanding proportions not only applies to numbers but also helps solve practical problems in geometry.
Using proportions, you can solve for unknowns in figures and predict relationships among sides of similar figures. Getting familiar with setting up and solving proportion equations is key to handling many math problems involving similar triangles.

Shadow Problems: Solving with Triangles

Shadow problems often involve using the principles of similar triangles, much like in our original exercise. They typically consist of large objects casting shadows and smaller, measurable objects such as a rod or pole also casting a shadow.
We use the shadows' lengths alongside the known height of one object to determine the unknown height of another. The inherent geometry concept here is that the Sun's rays, or the light source in general, causing the shadows, create similar triangles with the objects and their shadows.

• If two objects create shadows at the same time, the triangles they form with their shadows are similar.
• This means we can establish proportional relationships between the heights and the shadow lengths of the objects.

Practically, we start by identifying which triangles are similar. Once we set up the correct proportion to relate the triangle sides, we solve it using basic algebra or arithmetic operations. Shadow problems help translate abstract principles of geometry into practical tools for solving real-world measurements.

Geometry Concepts: The Power of Similar Triangles

Geometry often deals with shapes, sizes, and the properties of space. One powerful concept you frequently encounter is similar triangles. Identifying such triangles helps solve diverse problems due to their convenient properties.
The similarity of triangles means they have equal corresponding angles, and the sides around those angles are proportional. When working with geometry exercises like the shadow problem, this concept is particularly useful.

• For instance, right-angled triangles like those formed by shadows share angle properties that easily qualify them as similar if they share at least one angle.
• It's important to remember that the similarity ratio, or the scale factor, is crucial for converting measurements between different models of triangles.

A fundamental step in using similar triangles effectively is identifying the corresponding sides and angles in the triangles you compare. Once identified, you can apply proportions to solve for missing measurements. This makes similar triangles a cornerstone geometry concept that aids in seamlessly transitioning from theory to practical solution-finding.