Circles

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.
\( 41.81^{\circ} \)
\( 48.19^{\circ} \)
\( 33.69^{\circ} \)
\( 56.31^{\circ} \)

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.)
\( 60^{\circ} \)
\( 30^{\circ} \)
\( 90^{\circ} \)
\( 45^{\circ} \)

1103021602

Level: 
B
The side of an equilateral triangle is \( 6\,\mathrm{cm} \) long. Find the area of the annulus between the incircle and circumcircle of the given triangle. (See the picture.)
\( 9\pi\,\mathrm{cm}^2 \)
\( 6\pi\,\mathrm{cm}^2 \)
\( 12\pi\,\mathrm{cm}^2 \)
\( 8\pi\,\mathrm{cm}^2 \)

1103021604

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

1103021606

Level: 
B
In the rectangle \( ABCD \), \( a=6\,\mathrm{cm} \) and the radius of the circumcircle \( r=4\,\mathrm{cm} \) (see the picture). Find the measure of the angle between the diagonals of the rectangle. Round the result to two decimal places.
\( 82.82^{\circ} \)
\( 48.59^{\circ} \)
\( 97.18^{\circ} \)
\( 36.12^{\circ} \)

1103021608

Level: 
B
Consider a circle \( k \) with radius \( 2.5\,\mathrm{cm} \). 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 \( \sqrt{21}\,\mathrm{cm} \), and the length of \( DC \) is \( 4\,\mathrm{cm} \). What is the length of the shortest side of this quadrilateral? (See the picture.)
\( 2\,\mathrm{cm} \)
\( 3\,\mathrm{cm} \)
\( \sqrt5\,\mathrm{cm} \)
\( 2.5\,\mathrm{cm} \)

1103021609

Level: 
B
Points \( A \), \( B \) and \( C \) lie on the circle \( k \). The line segment \( AC \) is the diameter of the circle and the lines \( AC \) and \( BC \) contain the angle of \( 60^{\circ} \). Calculate the length of \( AC \) if the length of \( BC \) is \( 10\,\mathrm{cm} \). (See the picture.)
\( 20\,\mathrm{cm} \)
\( 5\sqrt3\,\mathrm{cm} \)
\( 5\,\mathrm{cm} \)
\( 2\sqrt3\,\mathrm{cm} \)

1103021611

Level: 
B
What is the length of the side of a regular pentagon circumscribed to a circle with radius of \( 9\,\mathrm{cm} \)? (See the picture.) Round the result to two decimal places.
\( 13.08\,\mathrm{cm} \)
\( 55.39\,\mathrm{cm} \)
\( 6.54\,\mathrm{cm} \)
\( 10.58\,\mathrm{cm} \)