B3. Vila Luxa is located at point $C$ on an island in the middle of a lake, while Hiša Krasna is at point $B$. The distance between points $A$ and $D$ is $10 \mathrm{~km}, \angle C B A=\angle B A C=28^{\circ}$.

A. What is the measure of $\angle C A D$?

B. How many kilometers is point $A$ from point $C$? Calculate to two decimal places.

C. What is the distance between Vila Luxa and Hiša Krasna?

![](https://cdn.mathpix.com/cropped/2024_06_07_2a119673c0ca3587e23fg-09.jpg?height=468&width=623&top_left_y=157&top_left_x=1296)

## 17th Mathematics Knowledge Competition for Vocational School Students

National Competition, April 22, 2017

## Solutions to Problems and Scoring for 1st and 2nd Year Students

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 one that

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

In the table, the correct answers to the multiple-choice questions are recorded. 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 | 7 | 8 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| C | C | $\mathrm{D}$ | $\mathrm{C}$ | $\mathrm{C}$ | $\mathrm{B}$ | $\mathrm{E}$ | $\mathrm{D}$ |
B3. Triangle $B C A$ is isosceles. Angle $A C B$ is $180^{\circ}-2 \cdot 28^{\circ}=124^{\circ}$, so angle $D C A$ is $56^{\circ}$. Since triangle $C D A$ is a right triangle, angle $C A D$ is $34^{\circ}$. The distance $|A C|$ can be calculated using $\cos 34^{\circ}=\frac{10}{|\mathrm{AC}|}$ or $|A C|=12.06 \mathrm{~km}$. Since triangle $B C A$ is isosceles, $|B C|=|A C|=12.06 \mathrm{~km}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_2a119673c0ca3587e23fg-13.jpg?height=58&width=1556&top_left_y=2144&top_left_x=296)

![](https://cdn.mathpix.com/cropped/2024_06_07_2a119673c0ca3587e23fg-13.jpg?height=69&width=1544&top_left_y=2213&top_left_x=313)

![](https://cdn.mathpix.com/cropped/2024_06_07_2a119673c0ca3587e23fg-13.jpg?height=57&width=1490&top_left_y=2267&top_left_x=366)

![](https://cdn.mathpix.com/cropped/2024_06_07_2a119673c0ca3587e23fg-13.jpg?height=52&width=1485&top_left_y=2321&top_left_x=368)

![](https://cdn.mathpix.com/cropped/2024_06_07_2a119673c0ca3587e23fg-13.jpg?height=66&width=1545&top_left_y=2377&top_left_x=310)