In the trapezoid $A B C D$ with bases $A B$ and $C D$, it is given that $|A D|=|C D|$, $|A B|=2|C D|$, $|B C|=24 \mathrm{~cm}$, and $|A C|=10 \mathrm{~cm}$.

Calculate the area of trapezoid $A B C D$.

(L. Růžičková)
Let $S$ be the midpoint of the base $AB$. From the problem statement, it follows that segments $AS, SB$, and $CD$ are congruent, so quadrilaterals $ASCD$ and $SBCD$ are parallelograms.

![](https://cdn.mathpix.com/cropped/2024_04_17_eab3d4ac54e010238b55g-3.jpg?height=383&width=1100&top_left_y=968&top_left_x=478)

Diagonal $SD$ divides parallelogram $ASCD$ into two congruent triangles. Similarly, diagonal $SC$ divides parallelogram $SBCD$ into two congruent triangles. The area of trapezoid $ABCD$ is therefore equal to three times the area of triangle $ASD$.

From the problem statement, we also know that segments $AD$ and $CD$ are congruent, so parallelogram $ASCD$ is a rhombus and its diagonals $SD$ and $AC$ intersect perpendicularly. Moreover, $|SD| = |BC| = 24 \, \text{cm}$ and $|AC| = 10 \, \text{cm}$, so the area of triangle $ASD$ is $\frac{1}{2} \cdot 24 \cdot 5 = 60 \, (\text{cm}^2)$.

The area of trapezoid $ABCD$ is therefore $3 \cdot 60 = 180 \, (\text{cm}^2)$.

Evaluation: 2 points for the relationship between the area of trapezoid $ABCD$ and the area of triangle $ASD$; 2 points for the perpendicularity of segments $SD$ and $AC$; 2 points for the calculation of the area and the quality of the commentary.