Circles

1103021612

Level: 
B
Consider two circles: the circle \( k \) with centre \( S_1 \) and radius \( 3\,\mathrm{cm} \), and the circle \( n \) with centre \( S_2 \) and radius \( 8\,\mathrm{cm} \). The distance between \( S_1 \) and \( S_2 \) is \( 22\,\mathrm{cm} \). Common internal tangents of the circles intersect at point \( A \). Calculate the distance of the point \( A \) from the centre \( S_1 \). (See the picture.)
\( 6\,\mathrm{cm} \)
\( 16\,\mathrm{cm} \)
\( 11\,\mathrm{cm} \)
\( 5\,\mathrm{cm} \)

1103021613

Level: 
B
A circle is inscribed in a rhombus \( ABCD \). The touching points of the circle and the rhombus divide each side into two parts that are \( 12\,\mathrm{dm} \) and \( 25\,\mathrm{dm} \) long. (See the picture.) Find the measure of the angle \( CAB \). Round the result to two decimal places.
\( 34.72^{\circ} \)
\( 43.85^{\circ} \)
\( 46.15^{\circ} \)
\( 23.14^{\circ} \)

1103077103

Level: 
B
The length of the shortest diagonal in a regular polygon is \( 8\,\mathrm{cm} \). The measure of the angle between this diagonal and the side of the polygon is \( 20^{\circ} \). Calculate the radius of a circle circumscribed about this polygon. Round the result to two decimal places.
\( 6.22\,\mathrm{cm} \)
\( 5.22\,\mathrm{cm} \)
\( 4.26\,\mathrm{cm} \)
\( 11.69\,\mathrm{cm} \)

1103077104

Level: 
B
Three equal circles, each of radius \( 6\,\mathrm{cm} \), touch each other as shown in the figure. Find the area of the region bounded by the circles. Round the result to one decimal place.
\( 5.8\,\mathrm{cm}^2 \)
\( 62.3\,\mathrm{cm}^2 \)
\( 6.2\,\mathrm{cm}^2 \)
\( 8.4\,\mathrm{cm}^2 \)

1103077105

Level: 
B
In a triangle \( ABC \), \( a=7\,\mathrm{cm} \), \( b=8\,\mathrm{cm} \), \( c=11\,\mathrm{cm} \). What is the radius of a circle circumscribed about this triangle? Round the result to two decimal places.
\( 5.51\,\mathrm{cm} \)
\( 6.11\,\mathrm{cm} \)
\( 4.92\,\mathrm{cm} \)
\( 6.52\,\mathrm{cm} \)

1103077106

Level: 
B
Let an equilateral triangle have a side of length of \( 10\,\mathrm{cm} \). Suppose there is a circular sector inside the triangle that has the centre at one of the vertices of the triangle, and the arc touches the opposite side (see the picture). Calculate the length of the arc of the sector. Round the result to two decimal places.
\( 9.07\,\mathrm{cm} \)
\( 8.62\,\mathrm{cm} \)
\( 8.93\,\mathrm{cm} \)
\( 9.05\,\mathrm{cm} \)

1103077107

Level: 
B
The figure shows an equilateral triangle whose side is \( 10\,\mathrm{cm} \) long. The circular sector inside the triangle has the centre at one of the vertices of the triangle, and the arc touches the opposite side. Find the ratio of the circumference of the sector to the perimeter of the triangle. Round the result to one decimal place.
\( 0.9 \)
\( 0.5 \)
\( 0.8 \)
\( 1.5 \)

1103077108

Level: 
B
The figure shows an equilateral triangle whose side is \( 10\,\mathrm{cm} \) long. The circular sector inside the triangle has the centre at one of the vertices of the triangle, and the arc touches the opposite side. Calculate the area of the sector. Round the result to one decimal place.
\( 39.3\,\mathrm{cm}^2 \)
\( 37.5\,\mathrm{cm}^2 \)
\( 14.4\,\mathrm{cm}^2 \)
\( 3.75\,\mathrm{cm}^2 \)