Question

In Exercises 83-100, solve for the indicated variable. Area of a Triangle Solve for \(h\) in \(A=\frac{1}{2} b h\).

Short answer

\(h = \frac{2A}{b}\)

Step-by-step solution

Understand the formula

The formula given is the general formula for the area of a triangle, where \(A\) is the area, \(b\) is the base, and \(h\) is the height. We are asked to solve for \(h\).

Rearrange the equation

To isolate \(h\), we first need to multiply both sides of the equation by 2. This will get rid of the fraction on the right hand side. It gives us: \(2A = b h\).

Divide by \(b\)

The final step to isolate \(h\) is to divide both sides of the equation by \(b\). This results in: \(h = \frac{2A}{b}\).

Key concepts

Area of a Triangle

Understanding the area of a triangle is fundamental in geometry. The formula to calculate the area of a triangle is given by \( A = \frac{1}{2} b h \), where \( A \) represents the area, \( b \) denotes the base, and \( h \) signifies the height of the triangle. This formula is derived from the broader concept that the area of a triangle is half the product of its base and height.

When visualizing this, imagine a rectangle and draw a diagonal line to create two triangles. Each of these triangles has exactly half the area of the rectangle, demonstrating why the triangle's area is half the product of height and base. This understanding is a stepping stone to solving for any of the variables in the formula.

Algebraic Manipulation

Algebraic manipulation involves rearranging an equation to solve for a specific variable. It's a skill that allows us to transform formulas based on the information we have or the information we seek. In this exercise, we are working with the area of a triangle and need to solve for the height \(h\).

To master algebraic manipulation, become comfortable with operations including multiplication, division, addition, and subtraction, and how they can be applied inversely to both sides of the equation to maintain equality. In our case, multiplying by 2 and dividing by \(b\) were the key steps to rearrange the formula. Being familiar with these inverse operations allows us to seamlessly switch between forms of an equation, such as turning a fraction into a whole number or isolating a variable.

Isolating Variables

Isolating a variable means to get the variable by itself on one side of the equation; it's the heart of solving algebraic equations. To isolate variables, we perform operations that 'undo' anything attached to the variable we're trying to solve for. For the area of a triangle, to isolate height \(h\), we removed the fraction by multiplying by 2 resulting in \(2A = bh\).

Then, we divided both sides by the base \(b\), because \(h\) was multiplied by \(b\), and we want to reverse this operation. As a result, we obtain \(h = \frac{2A}{b}\), effectively isolating \(h\). This step-by-step approach is crucial when dealing with complex formulas or when variables we aim to solve for are intertwined within an equation. Developing proficiency in isolating variables is a valuable mathematical skill in a variety of problem-solving scenarios.