5. The side $AD$ of trapezoid $ABCD$ is perpendicular to the bases $AB$ and $CD$. The lengths of the sides of this trapezoid are: $|AB|=2 \text{ cm}, |CD|=8 \text{ cm}$, and $|AD|=10 \text{ cm}$. Calculate the distance from the intersection of the diagonals $S$ to the base $AB$.

## You have 120 minutes to solve the problems.

Solve each problem independently on the provided paper, with each problem on a separate page. Do not sign the sheets; write only your code. Write your answers in pen, clearly and neatly. Draw graphs with a pencil. If you solve a problem in multiple ways, clearly indicate which solution the evaluator should grade. If you make a mistake, strike it through.

Each problem will be graded on a scale of 0 to 6 points.

The National Competition Committee wishes you great success.

## 7th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools

National Competition, April 21, 2007

## Problems for 3rd Year Students
5. We draw an appropriate sketch of the trapezoid where $a=2, c=8$ and $d=10$. On the sketch, we mark the intersection of the diagonals with $S$, the distance from the intersection to side $a$ with $x$, and the distance from the intersection to side $d$ with $y$. From similar triangles $\triangle A E S \approx \triangle A D C$ and $\triangle F B S \approx \triangle A B D$, we express the ratio, for example, $y: c=x: d$ and $(a-y): a=x: d$. Therefore, $y: 8=x: 10$ and $(2-y): 2=x: 10$. From this, we calculate the distance from the intersection of the diagonals $x=2 \mathrm{~cm}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_cd02ddf507985fa73630g-09.jpg?height=591&width=494&top_left_y=227&top_left_x=1341)

The drawn sketch with the intersection $S$ and both distances $x$ and $y$ marked. 1 point  
Writing the similar triangles ($E$ on side $d$ and $F$ on side $a$): $\triangle A E S \approx \triangle A D C$ and $\triangle F B S \approx \triangle A B D$. $1+1$ point

Set up the ratios $y: 8=x: 10$ and $(2-y): 2=x: 10$. $1+1$ point  
Calculated distance $x=2 \mathrm{~cm}$. 1 point

## Third Year