Expressing the Trapezoid Area: Using X to Account for Base Differences

Given the trapezoid where the height is equal to the sum of the two bases.

It is known that the difference between the large base and the small base is 5

We will mark the small base with X

Express the area of the trapezoid using X

Video Solution

Solution Steps

00:00 Express the area of the trapezoid using X

00:03 We will use the formula for calculating the area of a trapezoid

00:07 (sum of bases) multiplied by height) divided by 2

00:25 The height equals the sum of bases according to the given data

00:41 We will use the shortened multiplication formulas to expand the brackets

00:58 And this is the solution to the question

Step-by-Step Solution

To solve this problem, we will find the area of the trapezoid using the given expressions for the bases and height.

Step 1: Determine the height of the trapezoid.

The height, $h$ , is given as the sum of the two bases: $h = X + (X + 5) = 2X + 5$

Step 2: Apply the formula for the area of a trapezoid.

The formula for the area of a trapezoid is: $\text{Area} = \frac{1}{2} \times (\text{small base} + \text{large base}) \times \text{height}$
Substitute the expressions for the bases and height: $\text{Area} = \frac{1}{2} \times (X + (X + 5)) \times (2X + 5)$
Further simplifying: $\text{Area} = \frac{1}{2} \times (2X + 5) \times (2X + 5)$
This becomes: $\text{Area} = \frac{1}{2} \times (2X + 5)^2$
Expand $(2X + 5)^2$ using the square of a binomial formula: $(2X + 5)^2 = 4X^2 + 20X + 25$
Thus the area simplifies to: $\text{Area} = \frac{1}{2} (4X^2 + 20X + 25)$

Therefore, the expression for the area of the trapezoid in terms of $X$ is $\frac{1}{2}(4X^2 + 20X + 25)$ .