Circles

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.
\( 82.82^{\circ} \)
\( 48.59^{\circ} \)
\( 97.18^{\circ} \)
\( 36.12^{\circ} \)

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

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

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.)
\( 9\pi\,\mathrm{cm}^2 \)
\( 6\pi\,\mathrm{cm}^2 \)
\( 12\pi\,\mathrm{cm}^2 \)
\( 8\pi\,\mathrm{cm}^2 \)

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 \( DBC \)?
\( 5\,\mathrm{cm} \)
\( 4\,\mathrm{cm} \)
\( 6\,\mathrm{cm} \)
\( 8\,\mathrm{cm} \)

1103021511

Level: 
B
An acute triangle \( ABC \) is inscribed in the circle of radius \( r=4\,\mathrm{cm} \). Determine the measure of the angle \( ACB \), if the length of side \( c \) is \( 6\,\mathrm{cm} \). Round the result to two decimal places. (See the picture.)
\( 48.59^{\circ} \)
\( 97.18^{\circ} \)
\( 24.30^{\circ} \)
\( 41.41^{\circ} \)

1103021510

Level: 
A
A regular nonagon \( ABCDEFGHI \) is inscribed in a circle. Calculate the measures of all interior angles of the quadrilateral \( ABEH \). (See the picture.)
\( \alpha=120^{\circ};\ \beta=100^{\circ};\ \gamma=60^{\circ};\ \delta=80^{\circ} \)
\( \alpha=100^{\circ};\ \beta=120^{\circ};\ \gamma=60^{\circ};\ \delta=80^{\circ} \)
\( \alpha=100^{\circ};\ \beta=100^{\circ};\ \gamma=80^{\circ};\ \delta=60^{\circ} \)
\( \alpha=110^{\circ};\ \beta=130^{\circ};\ \gamma=70^{\circ};\ \delta=50^{\circ} \)