5. A chord of a circle has a length of $a$ and is $\frac{a}{2}$ away from the center of the circle. Express the radius of the circle in terms of $a$ and calculate the size of the acute and obtuse central angles over the chord. Draw a sketch.

## You have 120 minutes to solve.

Solve each problem independently on the provided paper, with each problem on a separate page. Write only your code on the sheets, not your name. 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, cross it out.

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

The National Competition Committee wishes you great success.

## 9th Competition in Mathematics Knowledge for Students of Secondary Technical and Vocational Schools National Competition, April 18, 2009

## Problems for 3rd Year Students
5. Using the Pythagorean theorem in triangle ATS, we express $r^{2}=\left(\frac{a}{2}\right)^{2}+\left(\frac{a}{2}\right)^{2}=\frac{a^{2}}{2}$. Then we take the square root and rationalize $r=\frac{a \sqrt{2}}{2}$. Since $A T S$ is an isosceles right triangle, $\varphi=45^{\circ}$. (The angle can also be calculated using trigonometric functions in the right triangle $A T S$, specifically: $\tan \varphi=\frac{\frac{a}{2}}{\frac{a}{2}}=1$.) We calculate the central angle $\varphi=45^{\circ}$. Then we calculate the two inscribed angles. The acute inscribed angle over the chord $A B$ measures $45^{\circ}$, and the obtuse one $135^{\circ}$.

Correct sketch 1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-10.jpg?height=517&width=528&top_left_y=1369&top_left_x=1324)

Application of the Pythagorean theorem $r^{2}=\frac{a^{2}}{2}$ 1 point

Calculation $r=\frac{a \sqrt{2}}{2}$

1 point

Application of trigonometric functions $\tan \varphi=\frac{\frac{a}{2}}{\frac{a}{2}}=1$.

1 point

Calculation of the angle $\varphi=45^{\circ}$

1 point

Calculation of the inscribed angles $45^{\circ}$ and $135^{\circ}$

1 point

## Third Year