2. The diagonals of a quadrilateral are equal, and the lengths of its midlines are p and q. Find the area of the quadrilateral.
Solution. Since $K L, L M, N M$ and $K N$ are the midlines of the corresponding triangles (see the figure), then $K L=N M=\frac{1}{2} A C, K N=L M=\frac{1}{2} B D$. According to the condition $A C=B D$. Therefore, $K L M N$ is a rhombus, the area of which is equal to $\frac{1}{2} p q$.

![](https://cdn.mathpix.com/cropped/2024_05_06_6e2a86ce89428713a89bg-1.jpg?height=411&width=571&top_left_y=1365&top_left_x=640)

Let the required area be $S$. Then $S=S_{\triangle A B D}+S_{\triangle B C D}$. But $S_{\triangle A K N}=\frac{1}{4} S_{\triangle A B D}$. Similarly, $S_{\triangle C M L}=\frac{1}{4} S_{\triangle B C D}$. Therefore, $S_{\triangle A K N}+S_{\triangle C M L}=\frac{1}{4} S$. Similarly, we obtain that $S_{\triangle K B L}+S_{\triangle M D N}=\frac{1}{4} S$. Thus, the area outside the rhombus is $\frac{1}{2} S$. Hence, the area of the rhombus is also $\frac{1}{2} S$, from which $S=p q$.

Answer. $S=p q$