Problem 11 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The base length \(b\) of the triangle, given its area \(A\) and height \(h\), can be calculated as \(b = \frac{2A}{h}\).

Step 1: Setup

The formula to calculate the Area (A) of a triangle is given as \(A = \frac{1}{2} b h\), where \(b\) is the base length of the triangle and \(h\) is the height. This problem requires solving for \(b\).

Step 2: Rearrange the formula

To solve for \(b\), the formula has to be rearranged. The first step is to get rid of the \(\frac{1}{2}\) to leave \(b\) alone with \(A\) on one side of the equation. This is done by multiplying through by 2 to get rid of the fraction. The equation now becomes \(2A = b h\).

Step 3: Final rearrangement

Finally, to completely isolate \(b\), divide both sides of the equation by \(h\). This yields: \(b = \frac{2A}{h}\).

Key Concepts

Algebra

Algebra is like the language of mathematics. It uses symbols and letters alongside numbers to represent objects and the things that happen to them. In this exercise, we use algebraic manipulation to solve the equation for a particular variable, which is the base, \(b\), of a triangle. Algebraic manipulation often includes rearranging equations, adding or removing terms, and using operations like multiplication and division. Let's break it down:

• Identify what you need to solve for (in this case, \(b\))
• Rearrange the terms to get the unknown variable on one side of the equation
• Use mathematical operations to isolate the variable

Practicing algebra helps you solve real-world problems by making sense of unknowns and how they interact with known values.

Geometry

Geometry is the study of shapes, sizes, and the properties of space. A triangle, being one of the most fundamental geometric shapes, is key to many concepts in geometry. Understanding the properties of triangles helps us work out various mathematical problems. Some points to always consider when dealing with geometry involve:

• The basic properties of shapes like the triangle
• Understanding how dimensions such as base and height define a shape
• The relationship between different parts of a shape

In this exercise, we see how knowing the dimensions of a triangle leads us to solve for the base when the area and height are known, demonstrating geometry's practical use.

Area of a Triangle

The area of a triangle is one of the fundamental measurement problems in geometry. The formula \(A = \frac{1}{2} b h\) calculates the space inside the triangle. You multiply the base \(b\) of the triangle by the height \(h\) and then divide by 2. This formula originates from the idea of a rectangle, since a triangle is essentially half of a rectangle.When solving for the base \(b\) given the area, you rearrange this formula. It involves:

• Multiplying both sides by 2 to eliminate the fraction \(\frac{1}{2}\)
• Dividing by the height \(h\) to isolate \(b\)

Understanding how to manipulate this formula is crucial as it frequently appears in diverse applications from basic math to real-world engineering problems.

Variables

Variables are characters, often represented by symbols like \(x\), \(y\), or \(b\), that take the place of unknown values in equations. In mathematics, especially algebra, variables are used to generalize problems and make them easier to solve. In the exercise question, \(b\) acts as a variable for the base of the triangle. Here's how we deal with variables:

• Variables can be manipulated using algebraic rules to solve equations
• They help us simplify the representation of complex problems
• Through substitution and rearrangement, we can find unknown quantities

By treating \(b\) as a placeholder, we focus on how the area and height influence each other, learning how to express a single quantity—base—in terms of area and height.