Solving for the Lengths of Bases in a Trapezoid with a Given Area and Height

Introduction

The problem we are addressing involves finding the lengths of the bases of a trapezoid given its area and the height along with the ratio of its parallel sides. This exercise not only enhances our understanding of the trapezoid's area formula but also demonstrates how to apply basic algebra to solve geometric problems.

Problem Statement

The area of a trapezoid is 84 square centimeters, and its height is 8 centimeters. The parallel sides of the trapezoid are in the ratio 2:5. We need to find the lengths of the bases.

Solution

We start with the formula for the area of a trapezoid, which is given by:

Area (1/2) * (b_1 b_2) * h

Here, ( b_1 ) and ( b_2 ) are the lengths of the parallel sides (bases), and ( h ) is the height of the trapezoid.

Given Information

Area 84 The ratio of the bases ( b_1 : b_2 2:5 )

Assigning Variables

Let's assign variables to the bases based on the given ratio:

Let ( b_1 2x ) Let ( b_2 5x )

Substituting into the Area Formula

Substituting into the area formula, we get:

84 (1/2) * (2x 5x) * 8

This simplifies to:

84 (1/2) * 7x * 8

(84 28x
)

Solving for ( x ):

(x 84 / 28 3)

Calculating the Lengths of the Bases

Now, we can find the lengths of the bases:

(b_1 2x 2 * 3 6 text{ cm} ) (b_2 5x 5 * 3 15 text{ cm} )

The lengths of the bases of the trapezoid are 6 cm and 15 cm.

Generalization: Using the Average of Parallel Sides Method

There is another method to solve this problem, often referred to as the average of the parallel sides method.

The average of the parallel sides can be found by dividing the area by the height:

84 / 8 10.5

This means the average of the parallel sides is 10.5 cm.

Since the bases are in the ratio 2:5, the total parts are 7 (2 5).

Each part is therefore:

10.5 * 2 / 7 3 text{ cm}
)

Thus, the shorter base (b_1) is 2 parts, and the longer base (b_2) is 5 parts:

(b_1 2 * 3 6 text{ cm}) (b_2 5 * 3 15 text{ cm})

Both methods yield the same result, confirming that the lengths of the bases are 6 cm and 15 cm.

Conclusion

We have successfully solved the problem by applying the trapezoid area formula and basic algebra. The lengths of the bases are 6 cm and 15 cm, respectively. Understanding and utilizing these methods can help in solving similar geometric problems effectively.