A7. In the figure below, a quadrilateral $A B C D$ is drawn. The midpoint of side $A B$ is $M$. The four line segments $A M$, $B M$, $B C$, and $A D$ have a length of 8, and the line segments $D M$ and $C M$ have a length of 5.

How long is line segment $C D$?

Note: the picture is not drawn to scale.

![](https://cdn.mathpix.com/cropped/2024_04_17_9e808ca4ac6657f5611fg-2.jpg?height=254&width=643&top_left_y=1712&top_left_x=1226)
A) 3
B) $\frac{40}{13}$
C) $\frac{25}{8}$
D) $\frac{16}{5}$
E) $\frac{13}{4}$
A7. C) $\frac{25}{8}$ Note that angle $A M B$ is straight. Therefore, $\angle A M D + \angle D M C + \angle C M B = 180^{\circ}$. Note that $\angle C M B = \angle M D A$ because triangles $A M D$ and $B M C$ are equal (three equal sides). Therefore, $\angle D M C = 180^{\circ} - \angle A M D - \angle M D A = \angle D A M$, because the three angles of triangle $A M D$ sum up to 180 degrees. Triangles $D M C$ and $D A M$ are thus two isosceles triangles with the same vertex angle. Therefore, they are equal to each other up to a scaling factor. This means that $\frac{|C D|}{|D M|} = \frac{|D M|}{|A D|}$. The length of $C D$ is thus equal to $\frac{5}{8} \cdot 5 = \frac{25}{8}$.