To find the area of the trapezoid $ABCD$, we'll use the information given in the diagram.

Step 1: Analyze the Given Information

• $ABCD$ is a trapezoid with parallel sides $AB$ and $CD$.
• The height $DH$ is given as 4 units.
• The length of the base $AB$ is 7 units.
• To find the area, we need the length of the other base $CD$.

Step 2: Calculate the Length of Base $CD$

Since $ABCD$ is an isosceles trapezoid (assuming symmetry because $AB$ is parallel to $CD$), the height $DH$ divides the trapezoid into two right triangles and a rectangle. The point $H$ is directly below point $D$ on base $AB$.

The segment $AH$ on base $AB$ is equal to the segment $DH$, as they form a 90-degree angle. So: $AH = 4 \text{ units}$

The remaining part of $AB$ (from $H$ to $B$) is: $HB = 7 - AH = 7 - 4 = 3 \text{ units}$

Since $ABCD$ is isosceles, this means the segment $BC$ is equal to $AH$, which is also 4 units. Thus, $CD = HB + BC = 3 + 4 = 7 \text{ units}$.

Step 3: Find the Area of the Trapezoid

The formula for the area $A$ of a trapezoid is given by: $A = \frac{1}{2} \times ( \text{Base}_1 + \text{Base}_2 ) \times \text{Height}$ Substituting the values: $A = \frac{1}{2} \times (7 + 7) \times 4 = \frac{1}{2} \times 14 \times 4 = 28 \text{ square units}$

So, the area of the trapezoid $ABCD$ is 28 square units.

Would you like further details or have any questions?

Here are 8 questions related to this problem:

1. What is the significance of knowing that a trapezoid is isosceles?
2. How would you calculate the area if $AB$ and $CD$ were not equal?
3. What other methods can be used to find the area of a trapezoid?
4. How does the height influence the area of a trapezoid?
5. Can the area formula for a trapezoid be applied to any quadrilateral?
6. How would the area change if the height were doubled?
7. What would happen to the area if the bases were not parallel?
8. How do you determine the height in a trapezoid if it is not given?

Tip: When working with trapezoids, always ensure that the height is perpendicular to the bases, as this is crucial for calculating the area accurately.