Circles

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.)
\( \alpha=100^{\circ};\ \beta=80^{\circ};\ \gamma=80^{\circ};\ \delta=100^{\circ} \)
\( \alpha=110^{\circ};\ \beta=80^{\circ};\ \gamma=80^{\circ};\ \delta=90^{\circ} \)
\( \alpha=110^{\circ};\ \beta=70^{\circ};\ \gamma=70^{\circ};\ \delta=110^{\circ} \)
\( \alpha=120^{\circ};\ \beta=80^{\circ};\ \gamma=80^{\circ};\ \delta=120^{\circ} \)

2010012810

Level: 
A
In the triangle \( KLM \), \( k=10\,\mathrm{cm} \), \( l=8\,\mathrm{cm} \), \( m=12\,\mathrm{cm} \). Point \( N \) is the foot of the altitude from the vertex \( K \) (See the picture.) What is the radius of the circumcircle of the triangle \( KLN \)?
\( 6\,\mathrm{cm} \)
\( 5\,\mathrm{cm} \)
\( 7\,\mathrm{cm} \)
\( 8\,\mathrm{cm} \)

1003021607

Level: 
B
Consider a right-angled triangle \( ABC \) with the right angle at the vertex \( C \). Calculate the measure of the angle \( CAB \), if the side \( b=9\,\mathrm{cm} \) and the radius of the circumscribed circle \( r=6\,\mathrm{cm} \). Round the result to one decimal place.
\( 41.4^{\circ} \)
\( 48.6^{\circ} \)
\( 36.9^{\circ} \)
\( 48.2^{\circ} \)

1003077112

Level: 
B
The length of a circular arc with the central angle of \( 3.5 \) radians is \( 82\,\mathrm{cm} \). Calculate the radius of the corresponding circle. Round the result to two decimal places.
\( 23.43\,\mathrm{cm} \)
\( 287.00\,\mathrm{cm} \)
\( 1.59\,\mathrm{cm} \)
\( 4217.40\,\mathrm{cm} \)

1003077113

Level: 
B
The curved surface of a cone has an area of \( 4.15\,\mathrm{cm}^2 \). If we flatten it into a plain, we get a sector of a circle with the central angle of \( 126^{\circ} \). Calculate the volume of this cone. Round the result to two decimal places.
\( 0.88\,\mathrm{cm}^3 \)
\( 0.62\,\mathrm{cm}^3 \)
\( 0.15\,\mathrm{cm}^3 \)
\( 311.00\,\mathrm{cm}^3 \)

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.)
\( 48.59^{\circ} \)
\( 97.18^{\circ} \)
\( 24.30^{\circ} \)
\( 41.41^{\circ} \)