Triangles

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.
\( 14.63^{\circ} \)
\( 29.27^{\circ} \)
\( 30.37^{\circ} \)
\( 28.30^{\circ} \)

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.
\( 3.21\,\mathrm{cm} \)
\( 1.71\,\mathrm{cm} \)
\( 3.83\,\mathrm{cm} \)
\( 6.42\,\mathrm{cm} \)

1103021601

Level: 
B
The distance from the point \( V \) to the centre \( S \) of the circle \( k \) is \( 30\,\mathrm{cm} \). The radius of the circle is \( 15\,\mathrm{cm} \). From the point \( V \) two tangent lines to the circle \( k \) can be drawn. What is the measure of the angle between them? (See the picture.)
\( 60^{\circ} \)
\( 30^{\circ} \)
\( 90^{\circ} \)
\( 45^{\circ} \)

1103021513

Level: 
B
The distance of the chord \( AB \) from the centre of the circle is equal to \( 2/3 \) of its radius. Find the measure of the angle \( SAB \). (See the picture.) Round the result to two decimal places.
\( 41.81^{\circ} \)
\( 48.19^{\circ} \)
\( 33.69^{\circ} \)
\( 56.31^{\circ} \)

1103021512

Level: 
A
In the triangle \( ABC \), \( a=10\,\mathrm{cm} \), \( b=8\,\mathrm{cm} \), \( c=12\,\mathrm{cm} \). Point \( D \) is the foot of the altitude from the vertex \( C \) (see the picture.) What is the radius of the circumcircle of the triangle \( DBC \)?
\( 5\,\mathrm{cm} \)
\( 4\,\mathrm{cm} \)
\( 6\,\mathrm{cm} \)
\( 8\,\mathrm{cm} \)

1103077011

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

1103077008

Level: 
C
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 \( AC \).
\( 4\sqrt7\,\mathrm{cm} \)
\( 2\sqrt7\,\mathrm{cm} \)
\( 2\sqrt{13}\,\mathrm{cm} \)
\( 4\sqrt{13}\,\mathrm{cm} \)