Problem 52 - Algebra 1

Question

The area A of a trapezoid with parallel bases of lengths \(b_{1}\) and \(b_{2}\) and height \(h\) is \(A=\frac{1}{2} h\left(b_{1}+b_{2}\right)\) Find the area of a trapezoid whose height is 2 meters and whose bases are 6 meters and 10 meters. (GRAPH CANNOT COPY)

Step-by-Step Solution

Answer

The area of the trapezoid is 16 square meters.

Step 1: Understand the formula

We are given the formula for the area of a trapezoid: \(A=\frac{1}{2} h\left(b_{1}+b_{2}\right)\). The meaning of the symbols are as follows: \(A\) is the area, \(h\) is the height of the trapezoid, and \(b_{1}\) and \(b_{2}\) are the lengths of the two bases.

Step 2: Substitute the given values

We substitute the given values into the formula. The height \(h\) is 2 meters, base \(b_{1}\) is 6 meters, and base \(b_{2}\) is 10 meters. So we can write: \(A=\frac{1}{2} * 2\left(6+10\right)\)

Step 3: Simplify and solve

Simplify the equation to compute the area. \(A = 1*(16) = 16\)

Step 4: Write and check the final answer

The area of this trapezoid is 16 square meters. Check and ensure the answer makes sense in the context of the problem.

Key Concepts

Understanding Geometry Through Trapezoids

Geometry is the study of shapes, sizes, and the properties of space. Understanding the basics of geometry involves exploring various shapes, including trapezoids. Trapezoids are four-sided shapes (quadrilaterals) with at least one pair of parallel sides known as bases. In real life, trapezoids can be seen in bridges, roofs, and handbags.
The fundamental properties of trapezoids allow us to delve into more complex concepts such as area calculation. Learning about trapezoids aids in identifying relationships between different geometric figures and facilitates the problem-solving process in geometry.

Area Calculation of a Trapezoid

Calculating the area of a trapezoid is an important skill in geometry. The area represents the amount of space inside the trapezoid.
To find the area, we use the formula:

• \( A = \frac{1}{2} h(b_{1} + b_{2}) \)
• Here, \( h \) is the height, while \( b_{1} \) and \( b_{2} \) are the lengths of the parallel bases.

Substitute the trapezoid's dimensions into the formula to calculate its area. For instance, with a height of 2 meters and bases of 6 meters and 10 meters:

• First, add the lengths of the bases: \( 6 + 10 = 16 \)
• Multiply by the height: \( 2 \)
• Divide by 2: \( \frac{1}{2} \times 2 \times 16 = 16 \)

The area of this trapezoid is 16 square meters. This procedure offers a systematic way to solve such geometry problems, ensuring precision and accuracy.

Approaching Mathematical Problem-Solving

Mathematical problem-solving involves several stages, from understanding the problem to checking the solution. Let's break down the process using the trapezoid's area calculation as an example.

• First, understand the problem: Identify known values and what you need to find. In this case, it's the area with given dimensions.
• Develop a plan: Use appropriate formulas like the trapezoid area formula.
• Execute the plan: Substitute the given values into the formula and simplify.
• Review the solution: Ensure the answer makes sense in the context of the question.

By honing these steps, students enhance their ability to tackle complex mathematical tasks. It's not just about finding the solution but also understanding the logical journey taken to get there.