10) In a trapezoid $A B C D$, the larger base $A B$ is triple the smaller base $C D$. Indicated with $E$ the midpoint of the diagonal $B D$, what is the ratio between the area of triangle $C D E$ and the area of the trapezoid?
(A) $1 / 3$
(B) $1 / 6$
(C) $1 / 8$
(D) $1 / 12$
(E) cannot be determined from the given data

![](https://cdn.mathpix.com/cropped/2024_04_17_9ee229095b4334063cc2g-2.jpg?height=265&width=406&top_left_y=121&top_left_x=1024)
10. The answer is $(\mathbf{C})$.

Let $H$ be the height of the trapezoid and $h$ the height of the triangle $E D C$ on the base $D C$. Considering the line parallel to the bases of the trapezoid and passing through $E$, it follows from Thales' theorem that $h=\frac{1}{2} H$. The area of the triangle is then $\mathcal{A}(D C E)=\frac{1}{2}(h \cdot D C)=\frac{1}{4}(H \cdot D C)$ while the area of the trapezoid is equal to $\mathcal{A}(A B C D)=\frac{1}{2}(A B+D C) H$. Therefore, $\frac{\mathcal{A}(D C E)}{\mathcal{A}(A B C D)}=\frac{D C}{2(A B+D C)}=\frac{D C}{2(3 D C+D C)}=\frac{1}{8}$.

[Problem proposed by P. Negrini]