A

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: 
A
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.)
\( \alpha=120^{\circ};\ \beta=100^{\circ};\ \gamma=60^{\circ};\ \delta=80^{\circ} \)
\( \alpha=100^{\circ};\ \beta=120^{\circ};\ \gamma=60^{\circ};\ \delta=80^{\circ} \)
\( \alpha=100^{\circ};\ \beta=100^{\circ};\ \gamma=80^{\circ};\ \delta=60^{\circ} \)
\( \alpha=110^{\circ};\ \beta=130^{\circ};\ \gamma=70^{\circ};\ \delta=50^{\circ} \)

1103021509

Level: 
A
A regular dodecagon \( ABCDEFGHIJKL \) is inscribed in a circle. Find the measures of all interior angles of the quadrilateral \( ABHJ \). (See the picture.)
\( \alpha=120^{\circ};\ \beta=75^{\circ};\ \gamma=60^{\circ};\ \delta=105^{\circ} \)
\( \alpha=105^{\circ};\ \beta=60^{\circ};\ \gamma=75^{\circ};\ \delta=120^{\circ} \)
\( \alpha=120^{\circ};\ \beta=30^{\circ};\ \gamma=60^{\circ};\ \delta=105^{\circ} \)
\( \alpha=105^{\circ};\ \beta=75^{\circ};\ \gamma=75^{\circ};\ \delta=105^{\circ} \)

1003021508

Level: 
A
A triangle is inscribed in a circle. Its vertices divide the circle into three arcs whose lengths are in the ratio \( 2:4:9 \). Determine the measures of the interior angles of the triangle.
\( 24^{\circ};\ 48^{\circ};\ 108^{\circ} \)
\( 30^{\circ};\ 40^{\circ};\ 110^{\circ} \)
\( 48^{\circ};\ 15^{\circ};\ 117^{\circ} \)
\( 15^{\circ};\ 60^{\circ};\ 105^{\circ} \)

1103021506

Level: 
A
Points \( A \) and \( B \) divide the circle \( k \) into two arcs whose lengths are in the ratio \( 5:13 \). Point \( C \) is an interior point of the longer arc. What is the degree measure of the angle \( ACB \)?
\( 50^{\circ} \)
\( 40^{\circ} \)
\( 100^{\circ} \)
\( 20^{\circ} \)

1103021503

Level: 
A
Determine the measure of the angle contained by two line segments: the first joining numbers \( 7 \) and \( 1 \), and the second joining numbers \( 1 \) and \( 4 \), on a clock face. (See the picture.)
\( 45^{\circ} \)
\( 60^{\circ} \)
\( 30^{\circ} \)
\( 90^{\circ} \)