Polygons

2000005904

Level: 
C
Find the magnitude of the angle that the diagonals \(DB\) and \(CG\) make in the regular heptagon \(ABCDEFG\). (See the picture.)
\( 180^{\circ}-\left(\frac{360^{\circ}}{14} +3\cdot\frac{360^{\circ}}{14}\right)\)
\( 180^{\circ}-\left(\frac{360^{\circ}}{7} +3\cdot\frac{360^{\circ}}{7}\right)\)
\( 180^{\circ}-\frac{360^{\circ}}{14} +3\cdot\frac{360^{\circ}}{14}\)
\( 180^{\circ}-\left(\frac{360^{\circ}}{14} +4\cdot\frac{360^{\circ}}{14}\right)\)

2000005510

Level: 
A
The lengths of the sides of the rectangular garden are in the ratio \(3:4\). The line connecting the centers of adjacent sides is \(25\,\mathrm{m}\) long (see the picture). How long will it take for the owner to till the whole garden if he digs over \(1\,200\,\mathrm{dm}^2\) per hour?
\(100\) hours
\(50\) hours
\(30\) hours
\(40\) hours

2000005508

Level: 
C
A rectangle with sides \(3\,\mathrm{cm}\) and \(4\,\mathrm{cm}\) long is divided by one of its diagonals into two triangles. What is the distance of the centers of gravity of these two triangles?
\(\frac{5}{3}\,\mathrm{cm}\)
\(\frac{4}{3}\,\mathrm{cm}\)
\(\frac{10}{3}\,\mathrm{cm}\)
\(2\,\mathrm{cm}\)

2000005507

Level: 
B
We cut two triangles from the rectangular plate so that the resulting trapezoid has an area of \(30\,\mathrm{cm}^2\). One of its bases is twice as long as the other. What is the area of the two triangles that are cut off?
\(10\,\mathrm{cm}^2\)
\(20\,\mathrm{cm}^2\)
\(5\,\mathrm{cm}^2\)
\(8\,\mathrm{cm}^2\)

2000005504

Level: 
C
Let \(ABCD\) be an arbitrary convex quadrilateral and let’s denote by \(P\), \(Q\), \(R\), \(S\) the centers of the sides \(AB\), \(BC\), \(CD\), \(DA\) in that order. Then, what type of a quadrilateral is \(PQRS\)?
It may or may not be a parallelogram.
It is a rectangle.
It is a rectangle or a square.
It is not a parallelogram.