Problem 13 - Intermediate Algebra

Question

Find the perimeter of each figure. See Example 1. A trapezoid with parallel sides 10 in. and 15 in. long and the other two sides each 6 in. long

Step-by-Step Solution

Answer

The perimeter of the trapezoid is 37 inches.

Step 1: Understand the Perimeter Concept

The perimeter of a figure is the total length around the figure. For a trapezoid, we add up all its side lengths.

Step 2: Identify the Sides of the Trapezoid

Identify the lengths of all the sides of the trapezoid. We have two parallel sides: 10 inches and 15 inches. The other two sides are 6 inches each.

Step 3: Set Up the Perimeter Formula

The formula for the perimeter of a trapezoid is \[ P = a + b + c + d \] where \( a \), \( b \), \( c \), and \( d \) are the lengths of the sides.

Step 4: Substitute the Side Lengths into the Formula

Substitute the known values into the formula: \[ P = 10 + 15 + 6 + 6 \]

Step 5: Calculate the Perimeter

Add up all the lengths:\[ P = 10 + 15 + 6 + 6 = 37 \, \text{inches} \]

Key Concepts

Understanding a Trapezoid

A trapezoid is a fascinating shape in geometry. It is a quadrilateral, which means it has four sides. However, what makes it unique is that it has one pair of parallel sides. These parallel sides are essential as they help distinguish a trapezoid from other quadrilaterals, such as squares or rectangles.

In the exercise, the trapezoid had two parallel sides: 10 inches and 15 inches long. The remaining sides, which are not parallel, are each 6 inches long. Knowing these side lengths is crucial for calculating properties like the perimeter.

What is Perimeter?

The perimeter is a basic yet vital concept in geometry. It represents the total length around a two-dimensional shape, in this case, a trapezoid.

You can think of the perimeter as the distance you would travel if you were to "walk" around the edges of the shape. It is a linear measure, and hence the units are usually the same as those used for measuring the sides, like inches in this example. Calculating the perimeter involves adding up the lengths of all the sides of the shape.

Step-by-Step Solution

A step-by-step solution is essential for understanding how to calculate the perimeter precisely. In our example, each step breaks down the process into manageable parts, making it easier to follow.

• Step 1: First, we recognized that the perimeter involves summing the lengths of each side.
• Step 2: We identified all the trapezoid's side lengths correctly, which ensures accuracy as we proceed.
• Step 3: Setting up an appropriate formula helps guide our calculations.
• Step 4 and 5: Substituting and adding the known values provide the solution, which is 37 inches in this case.

Understanding each step ensures you can apply this process to any trapezoid you encounter.

Using the Formula for Perimeter

The formula for calculating the perimeter of a trapezoid is simple yet powerful. It is expressed as \[ P = a + b + c + d \]where each letter represents one of the side lengths.

In this specific exercise, substituting the values was straightforward:

• \( a = 10 \) inches
• \( b = 15 \) inches
• \( c = 6 \) inches
• \( d = 6 \) inches

Applying it yielded a perimeter of 37 inches.

Mastering this formula aids in finding perimeters quickly and efficiently for any trapezoid by simply knowing and plugging in the side measurements.