Circles

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

2000005902

Level: 
A
What is the measure of the angle made by two line-segments marked in the clock face (see the picture)? One segment is connecting points \(1\) and \(3\), second segment is connecting points \(5\) and \(3\).
\( 120^{\circ}\)
\( 135^{\circ}\)
\( 60^{\circ}\)
\( 240^{\circ}\)

2000005903

Level: 
A
What is the measure of the angle made by two line-segments marked in the clock face (see the picture)? One segment is connecting points \(7\) and \(1\), second segment is connecting points \(5\) and \(10\).
\(105^{\circ}\)
\(120^{\circ}\)
\(115^{\circ}\)
\(75^{\circ}\)

2010012801

Level: 
A
A triangle is inscribed in a circle. Its vertices divide the circle into three arcs whose lengths are in the ratio \( 3:4:5 \). Determine the measures of the interior angles of the triangle.
\( 45^{\circ};\ 60^{\circ};\ 75^{\circ} \)
\( 20^{\circ};\ 60^{\circ};\ 100^{\circ} \)
\( 20^{\circ};\ 40^{\circ};\ 120^{\circ} \)
\( 50^{\circ};\ 60^{\circ};\ 70^{\circ} \)

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 \( ACB \)?
\( 36^{\circ}\)
\( 72^{\circ}\)
\( 24^{\circ}\)
\( 45^{\circ}\)