Problem 44 - Intermediate Algebra

Question

Height of a Triangle. If the height of a triangle with a base of 8 inches is tripled, its area is increased by 96 square inches. Find the height of the triangle.

Step-by-Step Solution

Answer

The original height of the triangle is 12 inches.

Step 1: Identify the Known Values

The base of the triangle is given as 8 inches. Let the original height be denoted as \( h \). The height is then tripled, making it \( 3h \).

Step 2: Write the Formula for Triangle Area

The area of a triangle is calculated using the formula \( A = \frac{1}{2} \times \text{base} \times \text{height} \). Initially, this gives us the area as \( A_1 = \frac{1}{2} \times 8 \times h = 4h \).

Step 3: Determine the New Area

When the height is tripled, the area becomes \( A_2 = \frac{1}{2} \times 8 \times 3h = 12h \). According to the problem, this new area is 96 square inches more than the original area.

Step 4: Set Up the Equation

The problem states that \( A_2 = A_1 + 96 \). Substitute the expressions for \( A_1 \) and \( A_2 \) to form the equation: \( 12h = 4h + 96 \).

Step 5: Solve for \( h \)

Subtract \( 4h \) from both sides of the equation \( 12h = 4h + 96 \) to get \( 8h = 96 \). Then, divide both sides by 8 to isolate \( h \): \( h = 12 \).

Key Concepts

Area of a Triangle

The area of a triangle is a fundamental concept in geometry that describes the space within its three sides. Understanding how to calculate the area of a triangle is essential in many mathematical applications. The formula to find the area of a triangle is given by:

• \( A = \frac{1}{2} \times \text{base} \times \text{height} \).

This formula works for any triangle, regardless of its type (scalene, isosceles, or equilateral). The base can be any one of the triangle's sides, and the height is the perpendicular distance from the base to the opposite vertex.
Choosing the correct base and corresponding height is crucial for using this formula accurately. In the exercise example, the base is given as 8 inches, and the height initially is represented by \( h \). By applying the formula, the area becomes \( A_1 = 4h \) with the original height, showing the direct impact height and base have on the triangle's area.

Algebraic Equations

Algebraic equations allow us to solve unknowns and find specific variable values, making them very important in mathematics and problem-solving. We often set up an equation using known formulas and values from the problem, as we did in our triangle area problem.
In our specific problem, we derived the new area, \( A_2 \), after the height was tripled. This was expressed as \( A_2 = 12h \) for the triangle with height \( 3h \). Then, using the condition that the area was increased by 96 square inches:

• \( A_2 = A_1 + 96 \)

Substituting the expressions for \( A_1 \) and \( A_2 \), we set up the equation: \( 12h = 4h + 96 \). This equation is pivotal in solving for \( h \) because it holds all the relationship conditions described in the problem.
Simplifying and solving this equation by isolating \( h \) gives us the answer for the triangle's original height.

Problem Solving

Problem solving in mathematics involves a series of logical steps that lead to a solution. In this scenario, we first identify the known quantities and unknowns. Then, finding a way to express relationships using algebraic equations is key.
Begin by extracting information, such as the base of 8 inches and the tripling of the height into \( 3h \). Then, apply the area of the triangle formula to set up the expression for the area both before and after the height is changed.

• Create the relationship: The increase in area by 96 square inches gives us a target condition.
• Using algebra, solve for the unknown \( h \) based on the given condition.

Throughout this process, revisit what each step means in context to understand how it contributes to the solution. This method is not only about computation but also developing a deeper understanding of how to approach similar problems in the future.