Triangles

9000121705

Level: 
A
Consider an isosceles triangle \(ABC\) with sides \(AC\) and \(BC\) of equal length. The measure of the angle \( BAC\) is \(40^{\circ }\). \(X\) is the point of intersection between the line $AB$ and the line through the vertex \(C\) perpendicular to it. Find the measure of the angle \( BCX\).
\(50^{\circ }\)
\(80^{\circ }\)
\(100^{\circ }\)
\(40^{\circ }\)

1003021803

Level: 
B
The ladder is leaning against the wall of a house. Its length is \( 6 \) meters. How high does the ladder reach if the angle between it and the wall is \( 30^{\circ} \)? (See the picture.)
\( 3\sqrt3\,\mathrm{m} \)
\( 3\,\mathrm{m} \)
\( 6\,\mathrm{m} \)
\( \frac{\sqrt3}2\,\mathrm{m} \)

1003021809

Level: 
B
\( ABC \) is a right-angled triangle with the right angle at vertex \( C \), side \( b=10\,\mathrm{cm} \) and the altitude to the hypotenuse \( v_c=5\,\mathrm{cm} \). Find the measure of the angle \( BAC \).
\( 30^{\circ} \)
\( 45^{\circ} \)
\( 60^{\circ} \)
\( 90^{\circ} \)

1003021902

Level: 
B
What is the width 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=\( 2.54\,\mathrm{cm} \))
\( 50.92\,\mathrm{cm} \)
\( 20.05\,\mathrm{cm} \)
\( 11.28\,\mathrm{cm} \)
\( 28.65\,\mathrm{cm} \)

1003021905

Level: 
B
Calculate the height between two floors if you know that the number of stairs between the floors is \( 16 \), the slope of the staircase is \( 30^{\circ} \) and the depth of a stair is \( 25\,\mathrm{cm} \).
\( \frac{400}{\sqrt3}\,\mathrm{cm} \)
\( \frac{25}{\sqrt3}\,\mathrm{cm} \)
\( 200\,\mathrm{cm} \)
\( 400\,\mathrm{cm} \)

1003076808

Level: 
B
In a triangle \( ABC \) the measure of \( \measuredangle CAB \) is \( 45^{\circ} \) and the measure of \( \measuredangle CBA \) is \( 60^{\circ} \). The altitude to side \( AB \) is \( 1\,\mathrm{cm} \) long. Calculate the area of the triangle \( ABC \) in \(\mathrm{cm}^2 \).
\( \frac{\sqrt3+1}{2\sqrt3} \)
\( \frac{\sqrt3+1}{\sqrt3} \)
\( \frac{\sqrt3+1}{2} \)
\( \frac{\sqrt3+1}{4} \)

1003077006

Level: 
B
Giving a right-angled triangle, the hypotenuse is \( 50\,\mathrm{cm} \) long, the perimeter of the triangle is \( 12\,\mathrm{dm} \) and its area is \( 600\,\mathrm{cm}^2 \). Find the measures of all interior angles of the triangle.
\( 90^{\circ};\ 36.87^{\circ};\ 53.13^{\circ} \)
\( 90^{\circ};\ 30.96^{\circ};\ 59.04^{\circ} \)
\( 90^{\circ};\ 38.65^{\circ};\ 51.35^{\circ} \)
\( 90^{\circ};\ 33.13^{\circ};\ 56.87^{\circ} \)