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
2010012901
Level:
B
Consider a circle \( k \) with radius \( 5\,\mathrm{cm} \). 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 \( 8\,\mathrm{cm} \), and the length of \( DC \) is \( 5\,\mathrm{cm} \). Determine the length of side \( AD \). (See the picture.)
\(5 \sqrt{3}\,\mathrm{cm} \)
\( 8\,\mathrm{cm} \)
\( 10\,\mathrm{cm} \)
\(3 \sqrt{5}\,\mathrm{cm} \)
2010012810
Level:
A
In the triangle \( KLM \), \( k=10\,\mathrm{cm} \), \( l=8\,\mathrm{cm} \), \( m=12\,\mathrm{cm} \). Point \( N \) is the foot of the altitude from the vertex \( K \) (See the picture.) What is the radius of the circumcircle of the triangle \( KLN \)?
\( 6\,\mathrm{cm} \)
\( 5\,\mathrm{cm} \)
\( 7\,\mathrm{cm} \)
\( 8\,\mathrm{cm} \)
2010012809
Level:
B
Which of the following formulas expresses the area of a regular pentagon inscribed in a circle of radius \( r \)? (See the picture.)
\( \frac{5r^2\sin72^{\circ}}2\)
\( \frac{10r^2\sin72^{\circ}}2\)
\( \frac{5r^2\sin36^{\circ}}2\)
\( \frac{5r \sin36^{\circ}}2\)
2010012808
Level:
A
A regular nonagon \( ABCDEFGHI \) is inscribed in a circle. Calculate the measures of all interior angles of the quadrilateral \( BDGI \). (See the picture.)
\( \alpha=100^{\circ};\ \beta=80^{\circ};\ \gamma=80^{\circ};\ \delta=100^{\circ} \)
\( \alpha=110^{\circ};\ \beta=80^{\circ};\ \gamma=80^{\circ};\ \delta=90^{\circ} \)
\( \alpha=110^{\circ};\ \beta=70^{\circ};\ \gamma=70^{\circ};\ \delta=110^{\circ} \)
\( \alpha=120^{\circ};\ \beta=80^{\circ};\ \gamma=80^{\circ};\ \delta=120^{\circ} \)
2010012807
Level:
A
A regular dodecagon \( ABCDEFGHIJKL \) is inscribed in a circle. Find the measures of all interior angles of the quadrilateral \( BFIL \). (See the picture.)
\( \alpha=90^{\circ};\ \beta=75^{\circ};\ \gamma=90^{\circ};\ \delta=105^{\circ} \)
\( \alpha=90^{\circ};\ \beta=60^{\circ};\ \gamma=80^{\circ};\ \delta=130^{\circ} \)
\( \alpha=80^{\circ};\ \beta=75^{\circ};\ \gamma=90^{\circ};\ \delta=115^{\circ} \)
\( \alpha=90^{\circ};\ \beta=105^{\circ};\ \gamma=90^{\circ};\ \delta=105^{\circ} \)
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 \( ACB \)?
\( 36^{\circ}\)
\( 72^{\circ}\)
\( 24^{\circ}\)
\( 45^{\circ}\)
2010012805
Level:
A
Give the measure of the angle between two line segments: the first connecting numbers \( 6 \) and \( 11 \), the second connecting numbers \( 8\) and \( 2\) on a clock face. (See the picture.)
\( 75^{\circ}\)
\( 30^{\circ}\)
\( 60^{\circ}\)
\( 45^{\circ}\)
2010012804
Level:
A
Give the measure of the angle contained by two line segments with the endpoints at the numbers \( 6 \), \( 5 \) and \( 6 \), \( 9 \) on a clock face. (See the picture.)
\( 120^{\circ}\)
\( 135^{\circ}\)
\( 100^{\circ}\)
\( 60^{\circ}\)
2010012803
Level:
A
What is the measure of the angle contained by two line segments: the first joining numbers \( 6 \) and \( 10 \), and the second joining numbers \( 6 \) and \( 4 \), on a clock face? (See the picture.)
\( 90^{\circ}\)
\( 60^{\circ}\)
\( 85^{\circ}\)
\( 95^{\circ}\)