Problem 26 - Prealgebra

Question

FIND THE ERROR Carla and Tony are finding the length of \(\overline{A B}\) where \(\triangle A B C \sim \triangle D E F, B C=16\) feet, \(E F=12\) feet, and \(D E=18\) feet. Who is correct? Explain your reasoning. $$\begin{aligned}&\begin{array}{l}\text { Carla } \\\\\frac{16}{18}=\frac{12}{x}\end{array}\\\&x=13.5 \mathrm{ft}\end{aligned}$$ $$\begin{aligned}&\begin{array}{r}\text { Tony } \\\\\frac{16}{12}=\frac{x}{18}\end{array}\\\&x=24 \mathrm{ft}\end{aligned}$$

Step-by-Step Solution

Answer

Tony is correct; \( \overline{AB} = 24 \) feet after solving the correct proportion.

Step 1: Understand the Problem

We are given two similar triangles, \( \triangle ABC \sim \triangle DEF \), with side lengths \( BC = 16 \) feet, \( EF = 12 \) feet, and \( DE = 18 \) feet. We need to find the length of \( \overline{AB} \) and determine which reasoning is correct: Carla's or Tony's.

Step 2: Set Up the Proportion

Since the triangles are similar, the sides are in proportion. Using the given sides \( BC \) and \( EF \) which are corresponding sides, the proportion is \( \frac{BC}{EF} = \frac{AB}{DE} \). This translates to \( \frac{16}{12} = \frac{x}{18} \).

Step 3: Solve Tony's Proportion

Substituting the known values, we have \( \frac{16}{12} = \frac{x}{18} \). Cross-multiplying gives \( 16 \times 18 = 12 \times x \), which simplifies to \( 288 = 12x \). Solving for \( x \), we divide both sides by 12: \( x = \frac{288}{12} = 24 \) feet.

Step 4: Evaluate Carla's Proportion

Carla uses the proportion \( \frac{16}{18} = \frac{12}{x} \). Cross-multiplying leads to \( 16x = 216 \), so \( x = \frac{216}{16} = 13.5 \) feet. This setup is incorrect for finding \( \overline{AB} \) since \( BC \) and \( EF \) are corresponding sides, not parts of the same fraction with \( DE \).

Step 5: Identify the Correct Solution

Since the correct proportion based on similar triangles is \( \frac{16}{12} = \frac{x}{18} \), the correct value of \( x \) should be 24 feet as calculated from Tony's setup.

Key Concepts

Understanding Proportions

Proportions play a vital role in solving geometric problems, especially with similar figures. In simple terms, a proportion is an equation that states that two ratios are equal. When you have similar triangles, corresponding sides are proportional, meaning their lengths have a constant ratio.

• For example, if we know the sides of two similar triangles, we can set up a proportion to find an unknown side length of one triangle.
• This method of using proportions is essential when dealing with similar triangles, as it allows us to solve for missing measurements efficiently.

The ability to set up and solve proportions correctly is a fundamental skill in geometry and problem-solving scenarios, as highlighted in the textbook problem between Carla and Tony.

Basic Concepts of Geometry

Geometry is the branch of mathematics that deals with shapes, sizes, and the relative positions of figures. When you explore geometric figures, you encounter various concepts, such as congruency and similarity.
Understanding these relations allows you to solve problems involving angles, lines, and triangles.

• Triangles, in particular, are a foundation of geometry because they can form more complex shapes and structures.
• In this context, similar triangles, which have the same shape but not necessarily the same size, provide a rich area for applying geometrical principles.

For example, in our exercise, knowing the principles of triangle similarity helps us set the correct proportions to find missing side lengths.

Effective Problem Solving

Solving a problem effectively requires a clear understanding of the question, setting up appropriate equations, and verifying solutions. The exercise outlined demonstrates this through triangle similarity and proportion setup.

• The first step is understanding the relationship between similar triangles and identifying corresponding sides.
• Next, setting up the correct proportion equation is crucial. This involves identifying which sides correspond to each other.

Once proportions are set, solving the equation through techniques like cross-multiplication leads to finding the correct answers. Rechecking calculations ensures the chosen method and result are accurate, as seen with Carla's and Tony's different approaches.

Principle of Triangle Similarity

Triangle similarity is a vital principle in geometry, allowing us to determine the relationship between different triangles that have identical shapes. If two triangles are similar, their corresponding sides are proportional, and corresponding angles are equal.

• This means if you know certain sides and angles in one triangle, you can deduce additional dimensions in a similar triangle using proportionality.

For the given problem, recognizing that \( \triangle ABC \sim \triangle DEF \) means their corresponding sides can form a proportion, which helps in solving for the unknown side length \( \overline{AB} \). This understanding highlights why Tony's method is correct and why Carla's approach had an error.

Cross-Multiplication Technique

Cross-multiplication is a mathematical technique used to solve equations involving proportions. It simplifies the process of finding unknown variables by eliminating fractions.

• This technique involves multiplying the outer terms (extremes) and the inner terms (means) of the proportion and then equating them to form a simple equation.
• In the exercise, we use cross-multiplication to solve for \( x \), the side length in the proportion equation \( \frac{16}{12} = \frac{x}{18} \).

By cross-multiplying, we convert the equation to a straightforward algebraic equation \( 16 \times 18 = 12 \times x \), which we can solve to find \( x = 24 \) feet. This technique is efficient and accurate in solving for unknowns when dealing with proportion-based problems.