Triangles

1103021906

Level: 
C
The distance between the places \( A \) and \( C \) on a straight road is \( 300\,\mathrm{m} \). There is a balloon \( B \) above the road between the places \( A \) and \( C \). See the picture. The angles of elevation from the places \( A \) and \( C \) to the balloon \( B \) are \( 20^{\circ} \) and \( 40^{\circ} \) respectively. What is the height \( h \) of the balloon?
\( 76\,\mathrm{m} \)
\( 168\,\mathrm{m} \)
\( 488\,\mathrm{m} \)
\( 523\,\mathrm{m} \)

1003021905

Level: 
C
Calculate the height between two floors if you know that the number of stairs between the floors is \( 16 \), the slope of the staircase is \( 30^{\circ} \) and the depth of a stair is \( 25\,\mathrm{cm} \).
\( \frac{400}{\sqrt3}\,\mathrm{cm} \)
\( \frac{25}{\sqrt3}\,\mathrm{cm} \)
\( 200\,\mathrm{cm} \)
\( 400\,\mathrm{cm} \)

1103021904

Level: 
C
From the highest window of Orava Castle, the angles of depression to the banks of the Orava river are \( 60^{\circ} \) and \( 20^{\circ} \). The height of the window above the river is \( 50\,\mathrm{m} \). What is the width of the river?
\( 108.5\,\mathrm{m} \)
\( 137.4\,\mathrm{m} \)
\( 100.5\,\mathrm{m} \)
\( 125.4\,\mathrm{m} \)

1103021903

Level: 
C
An observer was watching an approaching plane flying at a height of \( 3000\,\mathrm{m} \) in a straight line with constant velocity. At the first moment of measurement the observer saw the plane to be at an angle of elevation of \( 25^{\circ} \). After \( 10 \) seconds the angle of elevation changed to \( 35^{\circ} \). What was the speed of the plane? Round the result to ones.
\( 215\,\mathrm{m}\cdot\mathrm{s}^{-1} \)
\( 2149\,\mathrm{m}\cdot\mathrm{s}^{-1} \)
\( 6576\,\mathrm{m}\cdot\mathrm{s}^{-1} \)
\( 658\,\mathrm{m}\cdot\mathrm{s}^{-1} \)

1003021902

Level: 
C
What is the width of a computer screen if the ratio of its width and height is \( 16:9 \) and the computer has \( 23 \)-inch monitor? Round the result to two decimal places. (\( 1 \) inch=\( 2.54\,\mathrm{cm} \))
\( 50.92\,\mathrm{cm} \)
\( 20.05\,\mathrm{cm} \)
\( 11.28\,\mathrm{cm} \)
\( 28.65\,\mathrm{cm} \)

1103077011

Level: 
B
Consider a triangle \( ABC \) with \( a=1\,\mathrm{cm} \) and \( b = \sqrt3\,\mathrm{cm} \). The angle opposite the longer side is double the angle opposite the shorter side. Find the area of the triangle.
\( \frac{\sqrt3}2\,\mathrm{cm}^2 \)
\( 2\sqrt3\,\mathrm{cm}^2 \)
\( \sqrt3\,\mathrm{cm}^2 \)
\( \frac{\sqrt3}4\,\mathrm{cm}^2 \)

1003077010

Level: 
B
In an isosceles triangle \( ABC \) the base \( AB \) has length \( 12\,\mathrm{cm} \). The altitude to the base \( v_c=8\,\mathrm{cm} \). Determine the length of the median drawn from a vertex at the base to the side.
\( \sqrt{97}\,\mathrm{cm} \)
\( \sqrt{93}\,\mathrm{cm} \)
\( \sqrt{87}\,\mathrm{cm} \)
\( \sqrt{83}\,\mathrm{cm} \)

1103077008

Level: 
B
Given a triangle \( ABC \), the length of the median from \( C \) is \( 9\,\mathrm{cm} \) and the length of the median from \( B \) is \( 6\,\mathrm{cm} \). Let \( T \) be the centroid, and \( S \) be the midpoint of \( AC \). The measure of the angle \( BTC \) is \( 120^{\circ} \). Find the length of the side \( AC \).
\( 4\sqrt7\,\mathrm{cm} \)
\( 2\sqrt7\,\mathrm{cm} \)
\( 2\sqrt{13}\,\mathrm{cm} \)
\( 4\sqrt{13}\,\mathrm{cm} \)