Circles

2010012806

Level: 
A
Points \( A \) and \( B \) divide the circle \( k \) into two arcs whose lengths are in the ratio \( 3:12 \). Point \( C \) is an interior point of the longer arc. What is the degree measure of the angle \( ACB \)?
\( 36^{\circ}\)
\( 72^{\circ}\)
\( 24^{\circ}\)
\( 45^{\circ}\)

2010012807

Level: 
A
A regular dodecagon \( ABCDEFGHIJKL \) is inscribed in a circle. Find the measures of all interior angles of the quadrilateral \( BFIL \). (See the picture.)
\( \alpha=90^{\circ};\ \beta=75^{\circ};\ \gamma=90^{\circ};\ \delta=105^{\circ} \)
\( \alpha=90^{\circ};\ \beta=60^{\circ};\ \gamma=80^{\circ};\ \delta=130^{\circ} \)
\( \alpha=80^{\circ};\ \beta=75^{\circ};\ \gamma=90^{\circ};\ \delta=115^{\circ} \)
\( \alpha=90^{\circ};\ \beta=105^{\circ};\ \gamma=90^{\circ};\ \delta=105^{\circ} \)

2010012808

Level: 
A
A regular nonagon \( ABCDEFGHI \) is inscribed in a circle. Calculate the measures of all interior angles of the quadrilateral \( BDGI \). (See the picture.)
\( \alpha=100^{\circ};\ \beta=80^{\circ};\ \gamma=80^{\circ};\ \delta=100^{\circ} \)
\( \alpha=110^{\circ};\ \beta=80^{\circ};\ \gamma=80^{\circ};\ \delta=90^{\circ} \)
\( \alpha=110^{\circ};\ \beta=70^{\circ};\ \gamma=70^{\circ};\ \delta=110^{\circ} \)
\( \alpha=120^{\circ};\ \beta=80^{\circ};\ \gamma=80^{\circ};\ \delta=120^{\circ} \)

2010018002

Level: 
A
In the figure the cut of a regular polygon with unspecified number of vertices is shown. The red angle is the central angle of the polygon, the blue angle is the interior angle of the polygon. Suppose we consider a regular polygon with the central angle of \(30^{\circ}\), then find the measure of the interior angle of this polygon.
\(150^{\circ}\)
\(180^{\circ}\)
\(90^{\circ}\)
\(210^{\circ}\)

9000035002

Level: 
A
A line segment of the length \(40\, \mathrm{cm}\) joins two points on a circle. The radius of the circle is \(30\, \mathrm{cm}\). An angle has the vertex in the center of the circle and the arms on the ends of the line segment. Find the size of this angle and round the result to the nearest degrees and minutes.
\(83^{\circ }37'\)
\(97^{\circ }10'\)
\(41^{\circ }48'\)
\(96^{\circ }22'\)

9000036105

Level: 
A
The side \(b\) in the triangle \(ABC\) is \(17\, \mathrm{cm}\) and the angle \(\beta \) is \(58^{\circ }\). Find the radius of the circle circumscribed to this triangle and round your answer to the nearest centimeters.
\(10\, \mathrm{cm}\)
\(8\, \mathrm{cm}\)
\(9\, \mathrm{cm}\)
\(11\, \mathrm{cm}\)