Area of a Trapezoid: Express using

Let's calculate the area of trapezoid $ABCD$:

The formula for the area of a trapezoid is:

$\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}$

In this trapezoid, we have:

• $\text{Base}_1 = AB = 5$ cm
• $\text{Base}_2 = DC = 3$ cm
• $\text{Height} = h$

Substituting these into the formula, we get:

$\text{Area} = \frac{1}{2} \times (5 + 3) \times h$

Simplify the calculation:

$\text{Area} = \frac{1}{2} \times 8 \times h = 4h$

Thus, the area of the trapezoid is $4h$ square centimeters.

To express the area of the trapezoid in terms of $X$, follow these steps:

• Step 1: Identify the given values for the trapezoid's dimensions. The top base $b_1$ is $X + 14$, the bottom base $b_2$ is $3X + 7$, and the height $h$ is $2X$.
• Step 2: Use the trapezoid area formula $A = \frac{1}{2} \times (b_1 + b_2) \times h$.
• Step 3: Compute the sum of the bases: $(X + 14) + (3X + 7) = X + 14 + 3X + 7 = 4X + 21$.
• Step 4: Calculate the area using the formula: $A = \frac{1}{2} \times (4X + 21) \times (2X)$.
• Step 5: Simplify: $A = \frac{1}{2} \times (4X + 21) \times 2X = (4X + 21) \times X$.
• Step 6: Simplify further by distributing: $A = 4X^2 + 21X$.

Thus, the area of the trapezoid expressed in terms of $X$ is $4X^2 + 21X$.

To determine the area of the trapezoid, we will use the formula for the area of a trapezoid:

$A = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}$

From the problem, the two bases are $2X$ and $3Y$. The height is $X$.

Substituting into the formula, we have:

$A = \frac{1}{2} \times (2X + 3Y) \times X$

Simplifying the expression inside the parenthesis gives:

$A = \frac{1}{2} \times (2X + 3Y) \times X = \frac{1}{2} \times (2X \times X + 3Y \times X)$

Distributing $X$ through the terms inside the parenthesis gives:

$A = \frac{1}{2} \times (2X^2 + 3XY)$

Continuing the simplification:

$A = \frac{1}{2} \times 2X^2 + \frac{1}{2} \times 3XY$

Which simplifies to:

$A = X^2 + 1.5XY$

Therefore, the area of the trapezoid is $X^2 + 1.5XY$ cm².

Through comparison, this expression matches the given choice: $x^2+1.5xy$ cm², which corresponds to choice $3$.

Thus, the correct area of the trapezoid is $x^2 + 1.5xy$ cm².