2000005501
Level:
A
Derive the formula for the area of a square using the length of its diagonal \(u\).
\( \frac{u^2}{2}\)
\( 4u^2\)
\( \frac{u^2}{4}\)
\( 2u^2\)
Polygons
2000003203
Level:
C
A deltoid is composed of two isosceles triangles that have a common base. See the picture. Find the measures of the deltoids interior angles.
\( \alpha=36^{\circ};~\beta=134^{\circ};~\gamma=56^{\circ};~\delta=134^{\circ}\)
\( \alpha=36^{\circ};~\beta=100^{\circ};~\gamma=56^{\circ};~\delta=100^{\circ}\)
\( \alpha=56^{\circ};~\beta=134^{\circ};~\gamma=56^{\circ};~\delta=134^{\circ}\)
\( \alpha=36^{\circ};~\beta=128^{\circ};~\gamma=56^{\circ};~\delta=128^{\circ}\)
1103077103
Level:
C
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.
\( 6.22\,\mathrm{cm} \)
\( 5.22\,\mathrm{cm} \)
\( 4.26\,\mathrm{cm} \)
\( 11.69\,\mathrm{cm} \)
1103021613
Level:
C
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.
\( 34.72^{\circ} \)
\( 43.85^{\circ} \)
\( 46.15^{\circ} \)
\( 23.14^{\circ} \)
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.
\( 13.08\,\mathrm{cm} \)
\( 55.39\,\mathrm{cm} \)
\( 6.54\,\mathrm{cm} \)
\( 10.58\,\mathrm{cm} \)
1103021610
Level:
B
A regular hexagon \( ABCDEF \) is inscribed in a circle with radius of \( 12\,\mathrm{cm} \). Calculate the length of its diagonal \( EC \). (See the picture.)
\( 12\sqrt3\,\mathrm{cm} \)
\( 12\,\mathrm{cm} \)
\( 6\sqrt3\,\mathrm{cm} \)
\( 24\,\mathrm{cm} \)
1103021608
Level:
C
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.)
\( 2\,\mathrm{cm} \)
\( 3\,\mathrm{cm} \)
\( \sqrt5\,\mathrm{cm} \)
\( 2.5\,\mathrm{cm} \)
1103021606
Level:
A
In the rectangle \( ABCD \), \( a=6\,\mathrm{cm} \) and the radius of the circumcircle \( r=4\,\mathrm{cm} \) (see the picture). Find the measure of the angle between the diagonals of the rectangle. Round the result to two decimal places.
\( 82.82^{\circ} \)
\( 48.59^{\circ} \)
\( 97.18^{\circ} \)
\( 36.12^{\circ} \)
1003021603
Level:
B
Which of the following formulas expresses the area of a regular decagon inscribed in a circle of radius \( r \)? (See the picture.)
\( 10r^2\sin18^{\circ}\cos18^{\circ} \)
\( 10r^2\sin36^{\circ}\cos36^{\circ} \)
\( 5r^2\sin36^{\circ} \)
\( 5r^2\sin18^{\circ} \)
1003085406
Level:
A
A diagonal of a rectangle has length of \( 30\,\mathrm{cm} \) and the perimeter of the rectangle is \( 84\,\mathrm{cm} \). Find the difference of the length and the width of the rectangle in centimeters.
\( 6 \)
\( 8 \)
\( 9 \)
\( 10 \)