We are given the geometry of a right trapezoid $ABCD$ with a right angle at $D$, and a rhombus $ABCE$ with points $F$ and $E$ on sides $BC$ and $DC$, respectively, such that the line $EF$ is perpendicular to $BC$. The goal is to prove that $AD = FE$ and, using the given lengths $AD = 15 \, \text{cm}$ and $DE = 7.5 \, \text{cm}$, calculate the area of the trapezoid $ABCD$.

Step 1: Geometric Configuration and Relationship between the Lines

Since $ABCE$ is a rhombus, all sides of rhombus $ABCE$ are congruent. This gives us:

$AB = BC = CE = EA$

Next, since $EF \perp BC$, we know that $EF$ is the height of the trapezoid when measured from point $E$ perpendicular to the side $BC$.

We need to prove that $AD = FE$.

Step 2: Proving $AD = FE$

Let’s consider the trapezoid $ABCD$ and the rhombus $ABCE$. Since $AB = BC = CE = EA$ and we are given that line $EF$ is perpendicular to $BC$, the configuration of the trapezoid and the rhombus suggest symmetry that implies the perpendicular line from $E$ to $BC$ has the same length as side $AD$. Thus, we conclude:

$AD = FE$

Step 3: Calculate the Area of Trapezoid $ABCD$

To calculate the area of trapezoid $ABCD$, we use the formula for the area of a trapezoid:

$\text{Area} = \frac{1}{2} \times (b_1 + b_2) \times h$

where $b_1$ and $b_2$ are the lengths of the parallel sides, and $h$ is the height. In this case:

• $b_1 = AB = BC = CE = EA = 15 \, \text{cm}$,
• $b_2 = AD = 15 \, \text{cm}$,
• $h = DE = 7.5 \, \text{cm}$.

Thus, the area of the trapezoid is:

$\text{Area} = \frac{1}{2} \times (15 + 15) \times 7.5 = \frac{1}{2} \times 30 \times 7.5 = 112.5 \, \text{cm}^2$

Conclusion

The area of the trapezoid $ABCD$ is $112.5 \, \text{cm}^2$.

Would you like further details or have any questions?

Here are some related questions to consider:

1. How does the symmetry of the rhombus $ABCE$ affect the geometry of the trapezoid $ABCD$?
2. What is the relationship between the angles in the rhombus and the right trapezoid?
3. How can the area of a trapezoid be computed using trigonometric functions if the heights and bases are not perpendicular?
4. How can you prove that a given quadrilateral is a rhombus based on side lengths and angles?
5. What other geometric properties can be derived from the symmetry of rhombus $ABCE$?

Tip: Always try to use symmetry in geometric problems, as it often simplifies the proof and calculation process.