10.3. Given a trapezoid $A B C D$ and a point $M$ on the lateral side $A B$, such that $D M \perp A B$. It turns out that $M C=C D$. Find the length of the upper base $B C$, if $A D=d$.
Answer: $\frac{d}{2}$. Solution. Let $N$ be the midpoint of the lower base $AD$. Then, by the property of the median of the right triangle $AMD$, we have: $MN=AN=ND=\frac{d}{2}$. Consider $\triangle MCN$ and $\triangle DCN$. They are equal by three sides. Then $\angle MCN=\angle DCN$ and therefore, in the isosceles triangle $MCD$, the bisector is the altitude, i.e., $CN \perp MD$. Thus, $AB \| CN$ (as perpendiculars to the line $MD$). Hence, $ABCN$ is a parallelogram. Therefore, $BC=AN=\frac{d}{2}$.