## Task 5 - 100935

A triangular pyramid with vertices $A, B, C, D$ and apex $D$ has edge lengths $A B=4$ $\mathrm{cm}, A C=3 \mathrm{~cm}, B C=5 \mathrm{~cm}, B D=12 \mathrm{~cm}, C D=13 \mathrm{~cm}$, and $\angle A B D$ is a right angle.

Calculate the volume $V$ of this pyramid.
Given that $A B^{2}+A C^{2}=B C^{2}$, by the converse of the Pythagorean theorem, the triangle $\triangle A B C$ is a right triangle, where $A B$ and $A C$ are its legs.

Thus, its area is $A=\frac{1}{2} A B \cdot A C=6 \mathrm{~cm}^{2}$. Since $\angle A B D=90^{\circ}$, the segment $B D$ is also the height from the apex $D$ to the base $A B C$. Therefore, the volume $V$ of the pyramid is exactly $V=\frac{1}{3} A \cdot B D=24 \mathrm{~cm}^{3}$.