Example 27. In a regular triangular pyramid, a plane is drawn through the side of the base, perpendicular to the opposite lateral edge. Let's find the area of the resulting section, if the base is equal to $a$, and the height of the pyramid is equal to $h$.
Solution. Let $SABC$ be the given pyramid and $[SO]$ its height (Fig. 31). Draw the apothem $SD$ and connect point $D$ with point $C$ and with point $M$ - the vertex of triangle

![](https://cdn.mathpix.com/cropped/2024_05_21_318d1c34bf6979fc9790g-048.jpg?height=560&width=671&top_left_y=1770&top_left_x=1041)

Fig. 31
$ABM$, which is the given section. From the right triangle $SOC$ we obtain:

$$
|SC|=\sqrt{\frac{3 h^{2}+a^{2}}{3}}
$$

Next, $|DM| \cdot|SC|=|SO| \cdot|DC|=2 S_{\triangle SDC}$,

from which

$$
|DM|=\frac{3 a h}{2 \sqrt{3 h^{2}+a^{2}}}
$$

Thus,

$$
S_{\triangle ABM}=\frac{1}{2}|AB| \cdot|DM|=\frac{3 a^{2} h}{4 \sqrt{a^{2}+3 h^{2}}}.
$$

Investigation. Since $S_{\triangle ABM}>0$, we need to select only those values of $S_{\triangle ABM}$ that satisfy the inequality

$$
\frac{3 a^{2} h}{4 \sqrt{a^{2}+3 h^{2}}}>0
$$

It seems simple. Since $a>0$ and $h>0$, the found value of $S_{\triangle ABM}$ satisfies the condition of the problem. However, this is not the case. Indeed, if the pyramid $SABC$ is very "short," the plane passing through the line $AB$ perpendicular to the line $SC$ may not intersect the segment $SC$. Let's determine the relationship between the parameters $a$ and $h$ for which the section exists. Consider $\triangle CSD$. We have:

$$
|CD|=\frac{a \sqrt{3}}{2},|OD|=\frac{1}{3}|CD|=\frac{a \sqrt{3}}{6} \text { and }|OC|=\frac{2}{3}|CD|=\frac{a \sqrt{3}}{3} \text {. }
$$

If $\widehat{CSD}=90^{\circ}$, then $h=|SO|=\sqrt{|OD| \cdot|OC|}=\frac{a \sqrt{6}}{6}$, in this case point $M$ coincides with vertex $S$ and the section does not exist. Thus, it is clear that if

$$
\left\{\begin{array}{l}
a>0, \\
h>0, \\
h>\frac{a \sqrt{6}}{6}
\end{array}\right.
$$

i.e., if

$$
\left\{\begin{array}{l}
a>0 \\
h>\frac{a \sqrt{6}}{6}
\end{array}\right.
$$

then $S_{\triangle ABM}=\frac{3 a^{2} h}{4 \sqrt{a^{2}+3 h^{2}}}$.

In conclusion to this section, it should be noted that the investigation of the solution in the context of the problem should not be confused with the investigation of the obtained analytical expression of the desired quantity. The set of admissible values of parameters by the context of the problem is a subset of the set of admissible values of parameters for the analytical expression of the desired quantity. For example, in Example 27, the set of values of $a$ and $h$ defined by the system of inequalities

$$
\left\{\begin{array}{l}
a>0 \\
h>\frac{a \sqrt{6}}{6}
\end{array}\right.
$$

is a subset of the set defined by the system of inequalities

$$
\left\{\begin{array}{l}
a \neq 0 \\
h>0
\end{array}\right.
$$

which is equivalent to the inequality $\frac{3 a^{2} h}{4 \sqrt{a^{2}+3 h^{2}}}>0$.

In some cases, the sets of admissible values of parameters coincide.

For example, in Example 26, the admissible values by the context of the problem are $0<k \leqslant \frac{1}{2}$, and for the analytical expression $\cos x=$ $=\frac{1 \pm \sqrt{1-2 k}}{2}$, the admissible values are also $0<k \leqslant \frac{1}{2}$.

## Chapter II

## PLANE GEOMETRY

## § 6. Dependencies between Elements of Polygons