Polygons

2000006004

Level: 
C
In the parallelogram \(ABCD\), the side \(AB\) is \(10\,\mathrm{cm}\) long, the diagonal \(AC\) measures \(15\,\mathrm{cm}\). The distance of the vertex \(D\) from the diagonal \(AC\) is \(2\,\mathrm{cm}\). What is the distance of the vertex \(D\) from the side \(AB\)?
\(3\,\mathrm{cm}\)
\(4\,\mathrm{cm}\)
\(5\,\mathrm{cm}\)
\(6\,\mathrm{cm}\)

2000006008

Level: 
C
The trapezoid \(KLMN\) has bases \(15\,\mathrm{cm}\) and \(10\,\mathrm{cm}\) long. The point \(T\) is any point of the longer base. The area of the triangle \(MNT\) is \(40\,\mathrm{cm}^2\). What is the area of the trapezoid \(KLMN\)?
\(100\,\mathrm{cm}^2\)
\(80\,\mathrm{cm}^2\)
\(120\,\mathrm{cm}^2\)
\(50\,\mathrm{cm}^2\)

2010012901

Level: 
C
Consider a circle \( k \) with radius \( 5\,\mathrm{cm} \). In the circle is inscribed a convex quadrilateral \( ABCD \) so that the diagonal \( AC \) is the diameter of the circle, the length of \( BC \) is \( 8\,\mathrm{cm} \), and the length of \( DC \) is \( 5\,\mathrm{cm} \). Determine the length of side \( AD \). (See the picture.)
\(5 \sqrt{3}\,\mathrm{cm} \)
\( 8\,\mathrm{cm} \)
\( 10\,\mathrm{cm} \)
\(3 \sqrt{5}\,\mathrm{cm} \)

2010015003

Level: 
C
\( ABCD \) is a rhombus, the measure of the angle \( DAB \) is \(70^{\circ}\) and the shorter diagonal \( u = 50\,\mathrm{cm} \). Determine the height \(v\) of the rhombus. Round the result to two decimal places.
\( 40.96\,\mathrm{cm} \)
\( 28.68\,\mathrm{cm} \)
\( 71.41\,\mathrm{cm} \)
\( 46.98\,\mathrm{cm} \)

2010018004

Level: 
C
A rectangle-shaped land has dimensions \(5 \times 8\,\mathrm{cm}\) on a map with scale \(1:500\). The owner increased the size of his land by buying some land from his neighbor. The new land has dimensions \(7\times 9\,\mathrm{cm}\) on the map. Find the actual increase of the perimeter of the land (i.e. find the increase in the length of the fence required to enclose the whole land). Give your answer in meters.
\(30\,\mathrm{m}\)
\(15\,\mathrm{m}\)
\(40\,\mathrm{m}\)
\(60\,\mathrm{m}\)

9000124502

Level: 
C
A rectangle-shaped land has dimensions \(3\times 5\, \mathrm{cm}\) on a map with scale \(1\colon 2\: 000\). The owner increased the size of his land by buying some land from his neighbor. The new land has dimensions \(4\times 5\, \mathrm{cm}\) on the map. Find the actual increase of the perimeter of the land (i.e. find the increase in the length of the fence required to enclose the whole land). Give your answer in meters.
\(40\, \mathrm{m}\)
\(20\, \mathrm{m}\)
\(80\, \mathrm{m}\)
\(10\, \mathrm{m}\)

9000150502

Level: 
C
Two hotels and a lake are in a satellite photo. The distance between the hotels is \(400\, \mathrm{m}\) which is \(4\, \mathrm{cm}\) in the photo. The area of the lake in the photo is \(30\, \mathrm{cm}^{2}\). Find the real area of the lake.
\(3\cdot 10^{5}\, \mathrm{m}^{2}\)
\(3\cdot 10^{1}\, \mathrm{m}^{2}\)
\(3\cdot 10^{3}\, \mathrm{m}^{2}\)
There is not enough information to solve this problem.