Triangles

1003076906

Level: 
C
The lengths of the sides in a triangle are \( a \), \( b \), \( c \) and the opposite angles are \( \alpha \), \( \beta \), \( \gamma \). Give the measure of \( \alpha \) if \( a^2 = b^2 + c^2 +bc \).
\( 120^{\circ} \)
\( 60^{\circ} \)
\( 30^{\circ} \)
\( 90^{\circ} \)

1003076909

Level: 
C
\( ABC \) is a triangle. Given \( |AB|=3\,\mathrm{cm} \), the measure of \( \measuredangle CAB \) is \( 75^{\circ} \), and the measure of \(\measuredangle ABC \) is \( 45^{\circ} \), calculate the length of the side \( AC \).
\( \sqrt6\,\mathrm{cm} \)
\( 3\sqrt2\,\mathrm{cm} \)
\( 2\sqrt3\,\mathrm{cm} \)
\( 3\frac{\sqrt3}{\sqrt2}\,\mathrm{cm} \)

1003077010

Level: 
C
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.
\( \sqrt{97}\,\mathrm{cm} \)
\( \sqrt{93}\,\mathrm{cm} \)
\( \sqrt{87}\,\mathrm{cm} \)
\( \sqrt{83}\,\mathrm{cm} \)

1103021604

Level: 
C
Calculate the radius of a circle inscribed into the rhombus \( ABCD \) if the length of its side is \( 10\,\mathrm{cm} \) and the measure of the angle \( DAB \) is \( 40^{\circ} \). (See the picture.) Round the result to two decimal places.
\( 3.21\,\mathrm{cm} \)
\( 1.71\,\mathrm{cm} \)
\( 3.83\,\mathrm{cm} \)
\( 6.42\,\mathrm{cm} \)

1103021605

Level: 
C
A circle of radius \( 22\,\mathrm{cm} \) is inscribed into the rhombus \( ABCD \). Calculate the measure of the angle \( CAB \) if the length of the rhombus side is \( 90\,\mathrm{cm} \). (See the picture.) Round the result to two decimal places.
\( 14.63^{\circ} \)
\( 29.27^{\circ} \)
\( 30.37^{\circ} \)
\( 28.30^{\circ} \)

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.
\( 215\,\mathrm{m}\cdot\mathrm{s}^{-1} \)
\( 2149\,\mathrm{m}\cdot\mathrm{s}^{-1} \)
\( 6576\,\mathrm{m}\cdot\mathrm{s}^{-1} \)
\( 658\,\mathrm{m}\cdot\mathrm{s}^{-1} \)

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?
\( 108.5\,\mathrm{m} \)
\( 137.4\,\mathrm{m} \)
\( 100.5\,\mathrm{m} \)
\( 125.4\,\mathrm{m} \)