Problem 9 - Introductory and Intermediate Algebra for College Students 4th

Question

A triangle has a base of 5 feet and an area of 20 square feet Find the triangle's height.

Step-by-Step Solution

Answer

The height of the triangle is 8 feet.

Step 1: Understand the Task

The exercise requires the finding of the height of a triangle when the base and area are given. This necessitates the repositioning of the formula for calculating the area of a triangle to find the height. This is possible as the area, \( A \), equals the product of half of the triangle's base, \( b \), and its height, \( h \) - written as \( A = \frac{1}{2} \times b \times h \).

Step 2: Solve for the Height of the Triangle

You have \( A = 20 \) square feet and \( b = 5 \) feet. Substituting these into the area formula \( A = \frac{1}{2} \times b \times h \) yields \( 20 = \frac{1}{2} \times 5 \times h \). Multiply through by 2 to remove the fraction in the equation: \( 40 = 5 \times h \). Arrange for \( h \) to find the height of the triangle: \( h = \frac{40}{5} \)

Step 3: Calculating the Result

After arranging for \( h \), divide 40 by 5 to find the height of the triangle: \( h = 8 \) feet. This is consequently the height of the triangle.

Key Concepts

Formula Manipulation

When solving problems involving geometric shapes such as triangles, formula manipulation is a crucial skill. Here, it involves using known mathematical formulas in various ways to find an unknown quantity. In this exercise, we're given the triangle's area and base and asked to find the height.
To manipulate the area formula, start with the common formula for the area of a triangle:

• The area is given by \( A = \frac{1}{2} \times b \times h \), where \( A \) is the area, \( b \) is the base, and \( h \) is the height.
• We need the height, so we rearrange the formula to solve for \( h \). Multiply both sides by 2 to get rid of the fraction: \( 2A = b \times h \).
• Next, divide by the base, \( b \), to isolate \( h \): \( h = \frac{2A}{b} \).

This manipulation allows us to calculate the height once we know the area and the base.

Geometric Measurements

Geometric measurements relate to the dimensions and properties of shapes. Here, we focus on measuring a triangle's height using its area and base. To perform accurate geometric calculations, understanding the relationship between a shape's different dimensions is essential. In triangles, the main relevant aspects include:

• The Base (\( b \)): The bottom side of the triangle, serving as a reference in calculations.
• The Height (\( h \)): A perpendicular line from the base to the tip (or vertex) of the triangle, crucial for calculating area.
• The Area (\( A \)): The space covered by the triangle, measured in square units.

To use these measurements correctly, substitute values into formulas appropriately and perform algebraic operations to solve for unknown elements, such as the triangle's height.

Algebraic Equations

Algebraic equations are mathematical statements where two expressions are set equal to each other, and variables are used to represent unknown values. In the context of solving for a triangle's height, algebraic equations are used to rearrange and solve the area formula: \( A = \frac{1}{2} \times b \times h \).
Once we substitute the given values into the equation, it becomes crucial to understand:

• How to operate with fractions and eliminate them to simplify the equation, as done by multiplying through by 2.
• How to solve for a variable by isolating it, such as by dividing both sides by the base.
• Insert the given values, \( 20 = \frac{1}{2} \times 5 \times h \), manipulate to get \( h = 8 \).

Through these steps, algebra simplifies our calculations, allowing us to find the triangle's height using its area formula.