Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve for the indicated variable. Area of a Triangle $$\text { Solve for } h: A=\frac{1}{2} b h$$

Step-by-Step Solution

Answer

The formula for \( h \) when the formula for the area of a triangle is re-arranged is: \( h = \frac{2A}{b} \).

Step 1: Analyze the Formula

We have the formula for the area of a triangle \( A=\frac{1}{2} bh \), where \( A \) is the area, \( h \) is the height, and \( b \) is the base. We want to solve for \( h \).

Step 2: Isolate \( h \)

First, we will multiply both sides of the equation by \( 2 \) to remove \( \frac{1}{2} \) from the right side of the equation. That gives us \( 2A = bh \)

Step 3: Solve for \( h \)

Next, we will divide both sides of the equation by \( b \) to isolate \( h \) on one side. This gives us the final formula for \( h \): \( h = \frac{2A}{b} \)

Key Concepts

Area of a Triangle

Understanding the area of a triangle is foundational in geometry. The area, denoted by the symbol A, represents the amount of space that is enclosed within the three sides of the triangle. The classic formula provided in the exercise, \( A=\frac{1}{2} bh \) tells us that the area is equal to one half of the base times the height. The base, represented by b, can be any one of the sides, and the height, h, is the perpendicular distance from the base to the opposite vertex.

When you are given two of these quantities, finding the third one is just a matter of manipulating the formula appropriately. Let's say we know the area and the base, and we aim to find the height. We need to perform algebraic operations that will rearrange the formula solving for height, as you’ll see in the following sections.

Isolating Variables

Isolating a variable means rearranging an equation so that a single variable stands alone on one side of the equal sign. This is a vital skill in algebra and is essential to solve for a specific value you're looking for. In our triangle area problem (A), the goal is to solve for the height (h).

This process involves: 1) eliminating fractions or coefficients that are attached to the variable of interest and 2) ensuring that the variable is not mixed with any other terms on its side of the equation. In our case, the height is initially tied to a fraction and to the other variable, b. By multiplying the entire equation by 2, we remove the fraction, leaving us with a simpler equation, \( 2A = bh \) where we can more easily solve for h. Then, dividing both sides by the base (b) isolates h on one side, giving us the height in terms of A and b: \( h = \frac{2A}{b} \) .

Algebraic Manipulation

Algebraic manipulation includes the various operations that can be done to equations, like adding, subtracting, multiplying, or dividing both sides by the same number to maintain equality but to form a more useful version of the equation. This skill allows you to shape the equation to better fit the problem you are trying to solve.

In the triangle exercise, we used algebraic manipulation to rearrange the original area formula. Starting with \( A=\frac{1}{2} bh \) and our operation goals being to first eliminate the fraction and then remove the base from the right side, we tailored the formula step by step. We multiplied by 2 across and divided by b subsequently. Through this process, we showcased algebra’s power to transform an equation and isolate a variable, facilitating the solution for a specific unknown.