B3. Chords $A B$ and $C D$ intersect at point $S$ and form circular segments such that $|S B|=|S D|=6 \mathrm{~cm}$ (see figure).

A. Write the measure of angle $\alpha$.

B. Calculate the length of chord $A B$ if $|A S|:|S B|=5: 2$.

C. How long is the circular arc between points $B$ and $D$? Use the approximation $\pi = 3.14$ and write the result in centimeters to one decimal place.

![](https://cdn.mathpix.com/cropped/2024_06_07_debf03cdc366cb9b1122g-09.jpg?height=569&width=777&top_left_y=178&top_left_x=1139)

D. Using the appropriate trigonometric function, calculate the distance from point $D$ to chord $A B$. Write the result in centimeters to two decimal places.

## 18th Mathematics Knowledge Competition for Vocational School Students

National Competition, April 21, 2018

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

A contestant who arrives at the solution by any correct method (even if the scoring guide does not provide for it) 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.

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 | B | $\mathrm{D}$ | $\mathrm{B}$ | $\mathrm{B}$ | $\mathrm{E}$ | $\mathrm{D}$ | $\mathrm{D}$ |
B3. The angle $\alpha$ and the angle $40^{\circ}$ in the figure are congruent angles, so $\alpha=40^{\circ}$.

It holds: $|A S|:|S B|=5: 2$ or $\frac{|A S|}{6}=\frac{5}{2}$, hence $|A S|=15 \mathrm{~cm} .|A B|=|A S|+|S B|=21 \mathrm{~cm}$. The circumference of the entire circle with a radius of $6 \mathrm{~cm}$ is $37.68 \mathrm{~cm}$. The arc length between points $B$ and $D$ is $\frac{37.68 \mathrm{~cm} \cdot 40^{\circ}}{360^{\circ}} \approx 4.2 \mathrm{~cm}$.

The distance from point $D$ to the line segment $A B$ is denoted by $x$ and we write: $\sin 40^{\circ}=\frac{x}{6}$. The solution to the equation is $x=3.86 \mathrm{~cm}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_debf03cdc366cb9b1122g-13.jpg?height=57&width=1553&top_left_y=1790&top_left_x=300)

![](https://cdn.mathpix.com/cropped/2024_06_07_debf03cdc366cb9b1122g-13.jpg?height=60&width=1545&top_left_y=1855&top_left_x=307)

![](https://cdn.mathpix.com/cropped/2024_06_07_debf03cdc366cb9b1122g-13.jpg?height=52&width=1494&top_left_y=1913&top_left_x=361)

C. The calculated circumference of the circle with a radius of $6 \mathrm{~cm}$ is $37.68 \mathrm{~cm} . \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots . \ldots$ t

The calculated length of the arc between points $B$ and $D$ is $4.2 \mathrm{~cm} . \ldots \ldots \ldots \ldots . \mathbf{1 t}$

![](https://cdn.mathpix.com/cropped/2024_06_07_debf03cdc366cb9b1122g-13.jpg?height=66&width=1556&top_left_y=2097&top_left_x=296)

The calculated distance from point $D$ to the line segment $A B 3.86 \mathrm{~cm} \ldots \ldots \ldots \ldots \ldots \ldots \ldots .1 \mathbf{t}$