Problem 221 - Elementary Algebra

Question

In the following exercises, solve using triangle properties. What is the base of a triangle with area 207 square inches and height 18 inches?

Step-by-Step Solution

Answer

23 inches

Step 1: Recall the Area Formula of a Triangle

The formula for the area of a triangle is given by \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \].

Step 2: Write the Given Values

The problem gives the area of the triangle as 207 square inches and the height as 18 inches. Let's denote these values as: \[ \text{Area} = 207 \text{ square inches},\ \text{height} = 18 \text{ inches} \].

Step 3: Substitute the Given Values into the Formula

Substitute the given values into the area formula to set up the equation: \[ 207 = \frac{1}{2} \times \text{base} \times 18 \].

Step 4: Solve for the Base

Rearrange the equation to isolate the base. First, simplify the right-hand side: \[ 207 = \frac{1}{2} \times \text{base} \times 18 \] can be written as \[ 207 = 9 \times \text{base} \] (since \[ \frac{1}{2} \times 18 = 9 \]).Now, solve for the base: \[ \text{base} = \frac{207}{9} = 23 \text{ inches} \].

Key Concepts

Area of a triangle

Understanding the area of a triangle is crucial for solving many geometry problems. The formula for the area of a triangle is given by: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]. This formula tells us how to calculate the area using the base and the height of the triangle.
The base is one of the triangle's sides, and the height is the perpendicular distance from this side to the opposite vertex.
To solve the problem given in the exercise: we are given the area (207 square inches) and the height (18 inches). By rearranging the formula, we can find the missing base.
Solving geometric problems often requires rearranging formulas to isolate the unknown variable. In this case, we isolate the base to find its length.
Once the base is isolated, we substitute the known values and solve for the base.

Geometry

Geometry is the branch of mathematics that deals with shapes, sizes, and properties of space.
Triangles are fundamental shapes in geometry, and understanding their properties is essential. Triangles have three sides, three angles, and their angles always add up to 180 degrees.
In our exercise, we focus on calculating the area of a triangle using its base and height. This concept is helpful for various real-world applications, such as measuring land or understanding architectural designs.
By mastering the properties of triangles, you become adept at solving diverse geometric problems. Knowing properties like the sum of angles, congruency, and similarity further helps in analyzing complex shapes by breaking them down into simpler ones.
This foundational understanding enables you to connect different geometric concepts and apply them in practical situations.

Basic Algebra

Basic algebra involves manipulating equations and expressions to find unknown values.
In the given exercise, we used algebra to isolate the base of the triangle from the area formula. Let's break down the algebraic steps:

• First, we start with the equation from the area formula: \[ 207 = \frac{1}{2} \times \text{base} \times 18 \].
• Next, we simplify the right-hand side: \[ 207 = 9 \times \text{base} \] (since \[ \frac{1}{2} \times 18 = 9 \]).
• Finally, solve for the base by dividing both sides by 9: \[ \text{base} = \frac{207}{9} = 23 \text{ inches} \].

By following these steps, we used basic algebra to solve for the unknown base. This process of rearranging equations and isolating variables is a fundamental algebraic skill, which is essential for solving a wide range of mathematical problems.