Problem 15 - A Graphical Approach to College Algebra

Question

$$\begin{aligned} &\text {Solve each problem analytically, and support your solution}\\\ &\text {graphically.} \end{aligned}$$ Dimensions of a Puzzle Piece\(\quad\) A puzzle piece in the shape of a triangle has perimeter 30 centimeters. Two sides of the triangle are each twice as long as the shortest side. Find the length of the shortest side.

Step-by-Step Solution

Answer

The shortest side of the triangle is 6 cm.

Step 1: Define Variables

Let's define the shortest side of the triangle as \( x \) cm. According to the problem, the other two sides are each twice the shortest side's length, so each is \( 2x \) cm.

Step 2: Write the Perimeter Equation

The perimeter of the triangle is the sum of all its sides. We know from the problem that the total perimeter is 30 cm. Therefore, the equation for the perimeter can be written as:\[ x + 2x + 2x = 30 \]

Step 3: Simplify the Equation

Combine like terms in the perimeter equation. The terms \( 2x \) and \( 2x \) can be combined to give \( 4x \). Thus, the equation simplifies to:\[ x + 4x = 30 \]

Step 4: Solve for \( x \)

Combine the \( x \) terms in the equation to simplify it:\[ 5x = 30 \]Now, divide both sides of the equation by 5 to solve for \( x \):\[ x = \frac{30}{5} \]\[ x = 6 \]

Step 5: Interpret the Solution

The solution tells us that the shortest side of the triangle is \( 6 \) cm. Since the other two sides are twice this length, they are each \( 12 \) cm. Verify the solution by checking that the sum of these lengths equals the perimeter: \( 6 + 12 + 12 = 30 \) cm, which matches the given perimeter.

Key Concepts

Triangle Perimeter

The perimeter of a triangle is the total length around the triangle, which is simply the sum of all its sides. Understanding the concept of the triangle's perimeter is essential in solving many geometry problems, including this one. In this problem, knowing the perimeter helped set up an equation to find the unknown side lengths.For a triangle with sides labeled as \(a\), \(b\), and \(c\), the formula is:

• Perimeter = \(a + b + c\)

When given the perimeter, you can work backward to find the side lengths if some relationships among the sides are known. Here, the problem gives the perimeter and tells us two sides are each twice as long as the shortest side. This information is converted into an algebraic equation, making it easier to solve.

Algebraic Equations

An algebraic equation involves variables and constants, and solving them is a key skill in both algebra and geometry. In this exercise, the equation is derived from the problem's condition about the triangle's perimeter.We defined the shortest side as \(x\), and with the condition that the other sides each measure \(2x\), the equation appears as:

• \(x + 2x + 2x = 30\)

This step simplifies the problem by breaking it down into manageable numbers. By combining like terms—where all expressions with \(x\) are gathered together—the equation simplifies to \(5x = 30\). Solving this equation involves dividing both sides by 5 to isolate \(x\), resulting in \(x = 6\). This final value provides the length of the shortest side.

Problem Solving in Geometry

Approaching problem-solving in geometry often requires a strategic combination of algebraic skills and spatial understanding. This triangle problem illustrates a typical setup in geometry where relationships between side lengths are analyzed to find unknown dimensions. Here's a step-by-step problem-solving approach:

• Read and understand the problem statement carefully.
• Translate the geometric information into algebraic expressions or equations.
• Simplify and solve the equations to find the unknowns.
• Verify the solution by substituting back into the original conditions.

The key to successful problem-solving in geometry is practice, as it builds the required analytical and numerical skills. Verifying your results ensures accuracy and reinforces the understanding of underlying geometric principles.