Problem 15 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem analytically, and support your solution graphically. Dimensions of a Puzzle Piece 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 is 6 cm.

Step 1: Define Variables

Let the shortest side of the triangle be denoted as \( x \) centimeters. Consequently, each of the other two sides, which are twice as long as the shortest side, can be expressed as \( 2x \).

Step 2: Write the Perimeter Equation

The perimeter of the triangle, which is the sum of all its sides, is given by \( x + 2x + 2x = 30 \). Simplifying the equation gives \( 5x = 30 \).

Step 3: Solve for the Shortest Side

To find \( x \), the length of the shortest side, divide both sides of the equation by 5: \( x = \frac{30}{5} = 6 \).

Step 4: Verification and Visualization

To verify, substitute \( x = 6 \) back into the side lengths: the sides are \( 6, 12, \) and \( 12 \). The perimeter is \( 6 + 12 + 12 = 30 \), confirming the correctness. A triangle with sides \( 6 \), \( 12 \), and \( 12 \) is an isosceles triangle, and can be sketched to verify it visually.

Key Concepts

Triangle Perimeter

The perimeter of a triangle refers to the total length around the triangle, just like walking along each side. It’s the sum of the lengths of all its sides. In our puzzle piece problem, the perimeter is given as 30 centimeters. Knowing the perimeter helps us set up an equation to find unknown side lengths.
To form this equation, we add all the sides together. For this triangle, the sides are expressed in terms of the shortest side, with one side being the shortest and the other two being twice as long. By creating an equation as shown in the step-by-step solution, we can solve for unknown sides using simple algebra.

Algebraic Equations

Algebra allows us to find unknown quantities by creating equations. In this problem, we set up an equation to solve for the shortest side length by using the perimeter formula we derived.
We first let the shortest side be represented by the variable \( x \). Then, using the information that the other sides are twice this length, we write them as \( 2x \). Combining these expressions gives us the total perimeter equation: \( x + 2x + 2x = 30 \).
The key step here is simplifying the equation by combining like terms, resulting in \( 5x = 30 \). This linear equation can be easily solved by dividing both sides by 5, giving us \( x = 6 \), which tells us the shortest side is 6 centimeters.

Isosceles Triangle

An isosceles triangle is one with at least two sides of the same length. In our example, after solving the equation, the side lengths become 6 cm, 12 cm, and 12 cm. This confirms the triangle is isosceles because it has two equal sides.
Recognizing triangle types can be crucial in geometry problems, as it helps verify the correctness of a solution and can provide insights into potential properties like symmetry and angle measures.
The exercise highlights how understanding triangle types can assist in confirming arithmetic results through geometric visualization.

Mathematical Verification

Verification in mathematics means proving that your solution is correct. After determining the shortest side was 6 cm, it's essential to substitute back to verify our work.
Checking gives us side lengths of 6, 12, and 12 cm, and adding these confirms the perimeter is indeed 30 cm, which matches our original triangle perimeter requirement.
This process of going back and verifying every step ensures reliability and accuracy in problem-solving. Sketching the triangle can also help as a visual check that aligns with calculations, further solidifying our solution.