2010012902
Level:
B
Derive the formula for calculating the area of an equilateral triangle inscribed in a circle of radius \( r \).
\(\frac{3\sqrt{3}r^2}{4} \)
\(\frac{3\sqrt{3}r}{4} \)
\(\frac{3\sqrt{3}r^2}{2} \)
\(\frac{\sqrt{3}r^2}{4} \)
Circles
9000035002
Level:
B
A line segment of the length \(40\, \mathrm{cm}\)
joins two points on a circle. The radius of the circle is
\(30\, \mathrm{cm}\). An
angle has the vertex in the center of the circle and the arms on the ends of the line
segment. Find the size of this angle and round the result to the nearest degrees and
minutes.
\(83^{\circ }37'\)
\(97^{\circ }10'\)
\(41^{\circ }48'\)
\(96^{\circ }22'\)
9000045705
Level:
B
Find the formula for the radius \(r\)
of the circle circumscribed to the right triangle.
\(r = \frac{a}
{2\cdot \sin \alpha } \)
\(r = \frac{2a}
{\sin \alpha } \)
\(r = \frac{a}
{\sin 2\alpha } \)
\(r = \frac{a}
{\sin \alpha } \)
9000045706
Level:
B
Given a regular pentagon with the side
\(a\), find the
radius \(r\) of
the circle circumscribed to this pentagon.
\(r = \frac{a}
{2\cdot \cos 54^{\circ }}\)
\(r = \frac{2a}
{\cos 72^{\circ }}\)
\(r = \frac{2a}
{\cos 54^{\circ }}\)
\(r = \frac{a}
{2\cdot \cos 72^{\circ }}\)
9000045707
Level:
B
Given a regular pentagon with the side
\(a\), find the
radius \(\rho \) of
the circle inscribed to this pentagon.
\(\rho = \frac{a}
{2} \cdot \mathop{\mathrm{tg}}\nolimits 54^{\circ }\)
\(\rho = \frac{2a}
{\mathop{\mathrm{tg}}\nolimits 54^{\circ }}\)
\(\rho = \frac{a}
{2\cdot \mathop{\mathrm{tg}}\nolimits 54^{\circ }}\)
\(\rho = 2a\cdot \mathop{\mathrm{tg}}\nolimits 54^{\circ }\)
9000045708
Level:
B
Given a regular hexagon with the side
\(a\), find the
radius \(\rho \) of
the circle inscribed to this hexagon.
\(\rho = \frac{a}
{2\cdot \mathop{\mathrm{tg}}\nolimits 30^{\circ }}\)
\(\rho = 2a\cdot \mathop{\mathrm{tg}}\nolimits 30^{\circ }\)
\(\rho = \frac{2a}
{\mathop{\mathrm{tg}}\nolimits 30^{\circ }}\)
\(\rho = 2a\cdot \mathop{\mathrm{tg}}\nolimits 60^{\circ }\)
9000046405
Level:
B
A circle is circumscribed to the regular octagon. The perimeter of the octagon is
\(16\, \mathrm{cm}\). Find
the radius of the circle and round the result to two decimal places. (The regular
octagon is a polygon which has eight sides of equal length. The perimeter of the
octagon is the sum of the length of all eight sides.) Circle circumscribed to the regular octagon.
\(2.61\, \mathrm{cm}\)
\(1.08\, \mathrm{cm}\)
\(1.41\, \mathrm{cm}\)
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.
\( 6.64\,\mathrm{m}^2 \)
\( 0.57\,\mathrm{m}^2 \)
\( 0.35\,\mathrm{m}^2 \)
\( 1.52\,\mathrm{m}^2 \)
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.
\( 15.45\,\mathrm{m}^2 \)
\( 41.10\,\mathrm{m}^2 \)
\( 16.29\,\mathrm{m}^2 \)
\( 15.25\,\mathrm{m}^2 \)
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.
\( 56\,\% \)
\( 48\,\% \)
\( 62\,\% \)
\( 64\,\% \)