Triangles

2010015208

Level: 
A
In the triangle \( ABC \), \( \alpha=80^{\circ} \) and \( \gamma=30^{\circ} \) (see the picture). Determine the measure of the angle between the altitude to the side \( AC \) and the altitude to the side \( AB \).
\( 80^{\circ} \)
\(30^{\circ}\)
\(70^{\circ}\)
\(100^{\circ}\)

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=\( 2.54\,\mathrm{cm} \))
\( 28.64\,\mathrm{cm} \)
\(50.92\,\mathrm{cm} \)
\( 20.05\,\mathrm{cm} \)
\(11.28\,\mathrm{cm} \)

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.
\( 20\,\mathrm{cm} \)
\( \frac{20}{\sqrt3}\,\mathrm{cm} \)
\( 20\sqrt3\,\mathrm{cm} \)
\(15\,\mathrm{cm} \)

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.)
\( \frac{5\sqrt2}{2}\,\mathrm{m} \)
\( \frac{5}{2}\,\mathrm{m} \)
\( \frac{5\sqrt3}{2}\,\mathrm{m} \)
\( \frac{10}{\sqrt2}\,\mathrm{m} \)

2010015201

Level: 
A
Interior angles of a triangle \( ABC \) are in the ratio \( \alpha:\beta:\gamma=3:5:7 \). Calculate the measures of these angles.
\( \alpha=36^{\circ};\ \beta=60^{\circ};\ \gamma=84^{\circ} \)
\( \alpha=30^{\circ};\ \beta=50^{\circ};\ \gamma=70^{\circ} \)
\( \alpha=16.5^{\circ};\ \beta=30^{\circ};\ \gamma=133.5^{\circ} \)
\( \alpha=84^{\circ};\ \beta=60^{\circ};\ \gamma=36^{\circ} \)

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 \(\alpha\).
\( \frac{12}{13} \)
\( \frac{5}{13} \)
\( 22.62^{\circ} \)
\( 67.38^{\circ} \)