B4. Matej was standing on the lawn observing an old tree. Then he walked first $60 \mathrm{~m}$ north and then $50 \mathrm{~m}$ west and found that the tree was to the east of where he had arrived. He also found that he was as far from the tree as he was at the beginning of his journey. How many meters is he from the tree? Draw a sketch!

## 8th Mathematics Knowledge Competition for Vocational School Students  National Competition, April 19, 2008

## Solutions to Problems and Scoring

A competitor 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.


## Set A

The correct answers are recorded in the table. A correct answer by the competitor is scored 2 points, an incorrect answer $-\frac{1}{2}$ point, and a blank field in the table is scored 0 points.

| A1 | A2 | A3 | A4 | A5 | A6 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| B | C | D | B | B | D |
B4. Let's draw a sketch and mark Matej's initial position with $Z$, the final position with $K$, and the position of the tree with $D$. Since $Z D A$ is a right triangle, we can write the Pythagorean theorem: $(50+x)^{2}=60^{2}+x^{2}$. The equation has the solution $x=11$. Matej is $61 \mathrm{~m}$ away from the tree.

![](https://cdn.mathpix.com/cropped/2024_06_07_c432f7ef362881c6ef9bg-7.jpg?height=397&width=465&top_left_y=1732&top_left_x=844)

Scoring: Total: 7 points

A sketch is drawn, on which Matej's initial and final positions are marked, which are $60 \mathrm{~m}$ north and $50 \mathrm{~m}$ west of the initial position ............................................................................................................................

The tree's position is correctly marked ........................................................... $1 \mathrm{p}$

![](https://cdn.mathpix.com/cropped/2024_06_07_c432f7ef362881c6ef9bg-7.jpg?height=69&width=1651&top_left_y=2447&top_left_x=228)

![](https://cdn.mathpix.com/cropped/2024_06_07_c432f7ef362881c6ef9bg-7.jpg?height=49&width=1651&top_left_y=2511&top_left_x=228)

The answer is written, e.g., Matej is $61 \mathrm{~m}$ away from the tree. ........................................................... $1 \mathrm{p}$