4. From the midpoint $O$ of the height $SE$ of a right regular quadrilateral pyramid with apex $S$ and foot $E$ of the height, perpendiculars $OM$ are dropped to the lateral edge and $OK$ to the lateral face of the pyramid. If the lengths of these perpendiculars are $\overline{OM}=p$ and $\overline{OK}=q$, calculate the volume of the pyramid.
Solution. From the similarity of triangles $M O S$ and $EAS$, we get $\frac{H}{2}: p = s: \frac{a \sqrt{2}}{2}$, where $s$ is the length of the lateral edge, and $a$ is the length of the base edge. From triangles $SOK$ and $SFE$, we get $\frac{H}{2}: s = h: \frac{a}{2}$, where $H$ and $h$ are the height and the apothem of the pyramid, respectively. Substituting

$$
s^{2} = H^{2} + \frac{a^{2}}{2} \text{ and } h^{2} = H^{2} + \frac{a^{2}}{4}
$$

into the above proportions, we get

![](https://cdn.mathpix.com/cropped/2024_06_05_47c254b2217fdab00c02g-3.jpg?height=548&width=498&top_left_y=1646&top_left_x=991)

$$
\frac{a H}{4} \sqrt{2} = p \sqrt{H^{2} + \frac{a^{2}}{2}} \quad \text{ and } \quad \frac{a H}{4} = q \sqrt{H^{2} + \frac{a^{2}}{4}}.
$$

From the system, we get

$$
a^{2} = \frac{8 p^{2} q^{2}}{p^{2} - q^{2}} \quad \text{ and } \quad H^{2} = \frac{4 p^{2} q^{2}}{2 q^{2} - p^{2}},
$$

so

$$
V = \frac{1}{3} a^{2} H = \frac{16}{3} \frac{p^{3} q^{3}}{\left(p^{2} - q^{2}\right) \sqrt{2 q^{2} - p^{2}}}
$$

## III year