Circles

1103256903

Level: 
C
In isosceles triangle \( ABC \), \( |AB| = 8\,\mathrm{cm} \), \( |BC|=|AC| = 6\,\mathrm{cm} \). Determine what percentage of the triangle area is a circle that is inscribed in it. Round the result to full percentages.
\( 56\,\% \)
\( 48\,\% \)
\( 62\,\% \)
\( 64\,\% \)

1103256902

Level: 
C
The cucumber field has the shape of an isosceles right triangle. The length of its legs is \( 12\,\mathrm{m} \). Rotary sprinklers placed in its vertices have a reach of \( 6\,\mathrm{m} \). Find the area of the field that is not sprinkled with water. Round the result to two decimal places.
\( 15.45\,\mathrm{m}^2 \)
\( 41.10\,\mathrm{m}^2 \)
\( 16.29\,\mathrm{m}^2 \)
\( 15.25\,\mathrm{m}^2 \)

1103256901

Level: 
C
The farmer tied two goats on the meadow. The distance of the stakes \( K_1 \), \( K_2 \) to which the goats are tied is \( 5\,\mathrm{m} \) and the ropes have lengths of \( 3\,\mathrm{m} \) and \( 4\,\mathrm{m} \). What is the area of the grassland which is common for both goats? Round the result to two decimal places.
\( 6.64\,\mathrm{m}^2 \)
\( 0.57\,\mathrm{m}^2 \)
\( 0.35\,\mathrm{m}^2 \)
\( 1.52\,\mathrm{m}^2 \)

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.
\( 34.72^{\circ} \)
\( 43.85^{\circ} \)
\( 46.15^{\circ} \)
\( 23.14^{\circ} \)

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.)
\( 6\,\mathrm{cm} \)
\( 16\,\mathrm{cm} \)
\( 11\,\mathrm{cm} \)
\( 5\,\mathrm{cm} \)

1103021611

Level: 
B
What is the length of the side of a regular pentagon circumscribed to a circle with radius of \( 9\,\mathrm{cm} \)? (See the picture.) Round the result to two decimal places.
\( 13.08\,\mathrm{cm} \)
\( 55.39\,\mathrm{cm} \)
\( 6.54\,\mathrm{cm} \)
\( 10.58\,\mathrm{cm} \)

1103021609

Level: 
B
Points \( A \), \( B \) and \( C \) lie on the circle \( k \). The line segment \( AC \) is the diameter of the circle and the lines \( AC \) and \( BC \) contain the angle of \( 60^{\circ} \). Calculate the length of \( AC \) if the length of \( BC \) is \( 10\,\mathrm{cm} \). (See the picture.)
\( 20\,\mathrm{cm} \)
\( 5\sqrt3\,\mathrm{cm} \)
\( 5\,\mathrm{cm} \)
\( 2\sqrt3\,\mathrm{cm} \)

1103021608

Level: 
B
Consider a circle \( k \) with radius \( 2.5\,\mathrm{cm} \). In the circle is inscribed a convex quadrilateral \( ABCD \) so that the diagonal \( AC \) is the diameter of the circle, the length of \( BC \) is \( \sqrt{21}\,\mathrm{cm} \), and the length of \( DC \) is \( 4\,\mathrm{cm} \). What is the length of the shortest side of this quadrilateral? (See the picture.)
\( 2\,\mathrm{cm} \)
\( 3\,\mathrm{cm} \)
\( \sqrt5\,\mathrm{cm} \)
\( 2.5\,\mathrm{cm} \)

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.
\( 41.4^{\circ} \)
\( 48.6^{\circ} \)
\( 36.9^{\circ} \)
\( 48.2^{\circ} \)