B4. Cene, Ana, and Tine are sightseeing in Ljubljana. Their attention has been drawn to Congress Square, which has part of the road paved as shown in the picture. Tine is 60 meters away from the center of the square ($S$) and 45 meters away from the pavilion ($P$). Ana and Cene are standing 30 meters apart.

A What is the distance between Ana and Tine?

B How many meters is Cene away from the pavilion?

![](https://cdn.mathpix.com/cropped/2024_06_07_7876a7df18fe8c6add15g-3.jpg?height=568&width=740&top_left_y=1983&top_left_x=1155)

## 9th Mathematics Knowledge Competition for Vocational School Students  National Competition, April 18, 2009

## Solutions to the Problems and Scoring

A contestant who arrives at the solution by any correct method (even if not anticipated by the scoring guide) receives all possible points.

A correct method is considered any procedure that

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

$V$ in the table are the correct answers to the multiple-choice questions. Each correct answer is scored 2 points, an incorrect answer $-0.5$ points, and if the question is not answered, 0 points.

| 1 | 2 | 3 | 4 | 5 | 6 |
| :--- | :--- | :--- | :--- | :--- | :--- |
| $\mathrm{C}$ | $\mathrm{D}$ | $\mathrm{D}$ | $\mathrm{D}$ | $\mathrm{B}$ | $\mathrm{C}$ |
B4.

![](https://cdn.mathpix.com/cropped/2024_06_07_7876a7df18fe8c6add15g-6.jpg?height=434&width=551&top_left_y=1248&top_left_x=318)

Conclusion: Triangles $C A S$ and $P T S$ are similar. .1 point

Write the ratio $\frac{|A S|}{|A C|}=\frac{|T S|}{|T P|}$. 1 point

Calculate the length $|A S|=40 \mathrm{~m}$. .1 point

Answer: The distance between Ana and Tina is 100 m. .1 point

And the calculated distance $C S$ : 1 point

$$
\begin{aligned}
& |C S|=\sqrt{|A S|^{2}+|A C|^{2}} \\
& |C S|=\sqrt{40^{2}+30^{2}}=50 \mathrm{~m}
\end{aligned}
$$

Calculated distance $P S$ : 1 point

$$
\begin{aligned}
& |P S|=\sqrt{|T S|^{2}+|T P|^{2}} \\
& |P S|=\sqrt{60^{2}+45^{2}}=75 \mathrm{~m}
\end{aligned}
$$

Answer: Cene is 125 m away from the pavilion. 1 point