# Task 1. Maximum 15 points

In a small town, three main straight highways converge pairwise at points A, B, and C. The town has a circular ring road that also passes through points A, B, and C and is represented by a circle. The lengths of the highways inside the circle are 7, 8, and 9 kilometers. At the center of this circle is a local landmark - a television tower. Inside the city, there is a beautiful botanical garden, the territory of which is bordered by a live hedge of cypresses planted along a certain circle, so that the hedge touches each of the highways only once. The city authorities plan to build a suspended pedestrian bridge from the very center of the botanical garden directly to the television tower. Calculate the planned expenses for the construction of this bridge, if they amount to 1000 monetary units per 1 km of the distance between the center of the botanical garden and the television tower.
# Solution.

Mathematical formulation of the problem:

The sides of the triangle are 7, 8, and 9. Find the distance between the centers of the inscribed and circumscribed circles.

The sides of triangle \( \mathrm{ABC} \) are: \( \mathrm{AB}=7, \mathrm{BC}=8, \mathrm{AC}=9 \).

\( \mathrm{O} 2 \) - center of the inscribed circle, \( \mathrm{O} 1 \) - center of the circumscribed circle.

Using the cosine theorem to find \( \cos \angle \mathrm{C} \):

$$
49=81+64-2 \cdot 9 \cdot 8 \cos \angle C \rightarrow \cos \angle C=\frac{2}{3} \rightarrow \sin \angle C=\frac{\sqrt{5}}{3}
$$

\( S_{A B C}=\frac{9 \cdot 8 \cdot \sqrt{5}}{6}=12 \sqrt{5} \); radius of the circumscribed circle \( R=\frac{a b c}{4 S}=\frac{21}{2 \sqrt{5}} \);

radius of the inscribed circle \( r=\frac{2 s}{a+b+c}=\sqrt{5} \).

\( \cos \angle \mathrm{C}=\frac{2}{3} \rightarrow \operatorname{tg} \angle \frac{C}{2}=\frac{1}{\sqrt{5}} \).

From \( \triangle P O_{2} C \) we find \( \frac{r}{P C}=\operatorname{tg} \angle \frac{C}{2} \rightarrow \frac{\sqrt{5}}{P C}=\frac{1}{\sqrt{5}} \rightarrow P C=5 \).

$$
\begin{gathered}
O_{1} K \perp A C \rightarrow K C=\frac{9}{2} \rightarrow P K=\frac{1}{2} ; O_{1} K=\sqrt{R^{2}-K C^{2}}=\frac{3}{\sqrt{5}} \\
O_{1} K=\frac{3}{\sqrt{5}} \\
O_{1} S=\frac{2}{\sqrt{5}} \\
O_{1} O_{2}=\sqrt{\frac{1}{4}+\frac{4}{5}}=\frac{\sqrt{21}}{2 \sqrt{5}}
\end{gathered}
$$

Planned expenses:

\( 1000 \cdot \frac{\sqrt{21}}{2 \sqrt{5}}=\frac{500 \sqrt{21}}{\sqrt{5}} \sim 1024.7 \)

Answer: \( \frac{500 \sqrt{21}}{\sqrt{5}} \sim 1024.7 \).

## Criteria:

1) The radius of the circumscribed circle is calculated - 3 points
2) The radius of the inscribed circle is calculated - 3 points
3) The radii of the inscribed and circumscribed circles are calculated - 7 points
4) The radii of the inscribed and circumscribed circles are correctly calculated, the calculation of \( \mathrm{O}_{1} \mathrm{O}_{2} \) is correct, but there are minor errors, then 10 points
5) The solution is completely correct, the answer is correct, then 15 points
6) The solution process is incorrect (the mathematical formulation of the problem is incorrect) - 0 points