B3. The image shows the shape and dimensions of a metal plate in millimeters.

A. Draw a line segment in the plate with one endpoint at one of the plate's vertices so that a trapezoid is formed. Find all solutions.

B. How many mm is the external measurement $A$?

C. How many mm is the external measurement $B$?

D. Calculate the length of the slanted side $x$.

E. At what angle $\alpha$ do we cut the slanted side?

![](https://cdn.mathpix.com/cropped/2024_06_07_bbf95781e2fd378a6604g-05.jpg?height=537&width=693&top_left_y=1713&top_left_x=1224)
Write the result in degrees to two decimal places.

## Solutions and scoring for 1st and 2nd year students

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

For a correct method, each 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 | 7 | 8 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| B | B | B | B | A | B | D | E |
B3. To form a trapezoid, we draw one line segment as shown in the images.
![](https://cdn.mathpix.com/cropped/2024_06_07_bbf95781e2fd378a6604g-09.jpg?height=504&width=742&top_left_y=1916&top_left_x=696)

The external measurement $A$ is $A=27+8=35 \mathrm{~mm}$.

The external measurement $B$ is $B=8+8+6=22 \mathrm{~mm}$.

The slanted side $x$ represents the length of the hypotenuse of a right triangle with legs of lengths $k_{1}=22-14=8 \mathrm{~mm}$ and $k_{2}=27-12=15 \mathrm{~mm}$. The length of the hypotenuse $x=\sqrt{k_{1}^{2}+k_{2}^{2}}=\sqrt{8^{2}+15^{2}}=17 \mathrm{~mm}$.

The angle $\alpha$ is calculated using the tangent function, which defines the ratio of the lengths of the legs in the right triangle from the previous point: $\tan \alpha=\frac{k_{1}}{k_{2}}=\frac{8}{15}$. We get $\alpha=28.07^{\circ}$.

A. The drawn line segment. $1 \mathrm{t}$

B. The calculated length of the external measurement $A=35 \mathrm{~mm}$. $1 \mathbf{t}$

C. The calculated length of the external measurement $B=22 \mathrm{~mm}$. $1 \mathrm{t}$

D. The calculated length of the slanted side $x=17 \mathrm{~mm}$. $2 \mathrm{t}$

E. The calculated size of the angle $\alpha=28.07^{\circ}$. $2 \mathrm{t}$

Note: If the angle is not written to two decimal places, 1 point will be deducted.