9. The base of the pyramid $\mathrm{TABCD}$ is an isosceles trapezoid $\mathrm{ABCD}$, the midline of which is equal to $5 \sqrt{3}$. The ratio of the areas of the parts of the trapezoid $\mathrm{ABCD}$, into which it is divided by the midline, is $7: 13$. All lateral faces of the pyramid $\mathrm{TABCD}$ are inclined to the base plane at an angle of $30^{\circ}$. Find the volume of the pyramid TAKND, where points $\mathrm{K}$ and $\mathrm{N}$ are the midpoints of edges $\mathrm{TB}$ and $\mathrm{TC}$, respectively, and AD is the larger base of the trapezoid ABCD.
# Solution.

Let $TO$ be the height of the pyramid. Since all lateral faces are inclined to the base at the same angle, $O$ is the center of the circle inscribed in the base. Let $MP$ be the midline of the trapezoid, $\quad AD=a, BC=b. \quad$ According to the problem, we have $S_{MB CP}=7x, S_{\text{AMPD}}=13x, \quad \frac{7}{13}=\frac{b+5\sqrt{3}}{a+5\sqrt{3}}, a+b=10\sqrt{3}$, $a=8\sqrt{3}, b=2\sqrt{3}$. Since a circle can be inscribed in the trapezoid $ABCD$, then $AB+CD=a+b, \quad AB=CD=5\sqrt{3}$. We calculate the height of the trapezoid $h=\sqrt{AB^2-(a-b)^2/4}=4\sqrt{3}$. Draw a line through point $O$ perpendicular to the bases of the trapezoid and intersecting these bases at points $Q$ and $R, OR=h$. Since the lateral faces are inclined to the base plane at an angle of $30^\circ$, the height of the pyramid $TO=\frac{1}{2} QR \tan 30^\circ=2$.

Let $TF$ be the height of the pyramid TAKND, where $TF$ is the perpendicular dropped from point $T$ to the line $QS$, and $S$ is the midpoint of $KN$. We calculate the volume of the pyramid TAKND: $V_{\text{TAKND}}=\frac{1}{3} \cdot \frac{AD+KN}{2} \cdot QS \cdot TF$. The area of triangle $TQS$ can be calculated in two ways: $S_{\mathrm{TQS}}=\frac{QR \cdot TO}{4}, S_{\mathrm{TQS}}=\frac{QS \cdot TF}{2}, QS \cdot TF=\frac{QR \cdot TO}{2}$, $V_{\text{TAKND}}=\frac{1}{6} \cdot \frac{AD+KN}{2} \cdot QR \cdot TO$. From this, we get $V_{\text{TAKND}}=18$.

Answer: 18.
![](https://cdn.mathpix.com/cropped/2024_05_06_9e353629715b7c8c09e0g-05.jpg?height=1686&width=502&top_left_y=670&top_left_x=1436)

## First (online) stage of the academic competition "Step into the Future" for school students in the subject "Mathematics", autumn 2019, 11th grade

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