2010012902
Level:
B
Derive the formula for calculating the area of an equilateral triangle inscribed in a circle of radius \( r \).
\(\frac{3\sqrt{3}r^2}{4} \)
\(\frac{3\sqrt{3}r}{4} \)
\(\frac{3\sqrt{3}r^2}{2} \)
\(\frac{\sqrt{3}r^2}{4} \)
Triangles
2010015006
Level:
B
The figure shows a rectangular trapezium whose bases have lengths of \( 19\,\mathrm{cm} \) and \( 14\,\mathrm{cm} \), and the longer arm is \( 13\,\mathrm{cm} \) long. Calculate the sine of angle \(\alpha\).
\( \frac{12}{13} \)
\( \frac{5}{13} \)
\( 22.62^{\circ} \)
\( 67.38^{\circ} \)
2010015202
Level:
B
The ladder is leaning against the wall of a house. Its length is \( 5 \) meters. How high does the ladder reach if the angle between it and the wall is \( 45^{\circ} \)? (See the picture.)
\( \frac{5\sqrt2}{2}\,\mathrm{m} \)
\( \frac{5}{2}\,\mathrm{m} \)
\( \frac{5\sqrt3}{2}\,\mathrm{m} \)
\( \frac{10}{\sqrt2}\,\mathrm{m} \)
2010015203
Level:
B
In a triangle with interior angles \( 30^{\circ} \), \( 60^{\circ} \) and \( 90^{\circ} \), the shortest side is \( 10\,\mathrm{cm} \) long. Find the length of its longest side.
\( 20\,\mathrm{cm} \)
\( \frac{20}{\sqrt3}\,\mathrm{cm} \)
\( 20\sqrt3\,\mathrm{cm} \)
\(15\,\mathrm{cm} \)
2010015204
Level:
B
What is the height of a computer screen if the ratio of its width and height is \( 16:9 \) and the computer has \( 23 \)-inch monitor? Round the result to two decimal places. (\( 1 \) inch=\( 2.54\,\mathrm{cm} \))
\( 28.64\,\mathrm{cm} \)
\(50.92\,\mathrm{cm} \)
\( 20.05\,\mathrm{cm} \)
\(11.28\,\mathrm{cm} \)
2010015210
Level:
B
Given a right-angled triangle \( ABC \) with the right angle at vertex \( C \), calculate \( \cos\beta \) if \(\sin\alpha=\frac4{7} \).
\( \frac4{7} \)
\( \frac7{4} \)
\( \frac{\sqrt{33}}{7} \)
\( \frac7{\sqrt{33}} \)
2010015301
Level:
B
An artillery battery is placed on a cliff. From the edge of the cliff \( 200\,\mathrm{m} \) high, the angle of depression to the ship on the sea is \( 10^{\circ} \). What is the distance \( d \) (See the picture.) from the cliff to the ship? Round the result to two decimal places.
\( 1134.26\,\mathrm{m} \)
\( 1151.75\,\mathrm{m} \)
\( 35.27\,\mathrm{m} \)
\( 203.09\,\mathrm{m} \)
2010015302
Level:
B
The slope of a staircase is \(30^{\circ}\). What is the depth of a stair if its height is \( 15\,\mathrm{cm} \)? (See the picture.)
\(15\sqrt3\,\mathrm{cm} \)
\(30\,\mathrm{cm} \)
\( \frac{15}{\sqrt3}\,\mathrm{cm} \)
\( 28\,\mathrm{cm} \)
2010015307
Level:
B
The angle of elevation of a straight road is \(9^{\circ }\). The distance between two places measured along the road is \(2\, \mathrm{km}\). For these places, find the difference in altitudes, i.e. the vertical distance, and round the result to the nearest meter. (See the picture.)
\(313\, \mathrm{m}\)
\(1975\, \mathrm{m}\)
\(317\, \mathrm{m}\)
\(78\, \mathrm{m}\)
2010015308
Level:
B
A roof gable has the shape of an isosceles triangle with the base of \(14\, \mathrm{m}\) and the height \(6\,\mathrm{m}\). What is the angle between the roof and the horizontal direction? (See the picture.) Round your result to one decimal place.
\(40.6^{\circ}\)
\(49.4^{\circ}\)
\(59.0^{\circ}\)
\(31.0^{\circ}\)