Triangles

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 \( BAC \).
\( 30^{\circ} \)
\( 45^{\circ} \)
\( 60^{\circ} \)
\( 90^{\circ} \)

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 \(\mathrm{cm}^2 \).
\( \frac{\sqrt3+1}{2\sqrt3} \)
\( \frac{\sqrt3+1}{\sqrt3} \)
\( \frac{\sqrt3+1}{2} \)
\( \frac{\sqrt3+1}{4} \)

1003076906

Level: 
B
The lengths of the sides in a triangle are \( a \), \( b \), \( c \) and the opposite angles are \( \alpha \), \( \beta \), \( \gamma \). Give the measure of \( \alpha \) if \( a^2 = b^2 + c^2 +bc \).
\( 120^{\circ} \)
\( 60^{\circ} \)
\( 30^{\circ} \)
\( 90^{\circ} \)

1003076909

Level: 
B
\( ABC \) is a triangle. Given \( |AB|=3\,\mathrm{cm} \), the measure of \( \measuredangle CAB \) is \( 75^{\circ} \), and the measure of \(\measuredangle ABC \) is \( 45^{\circ} \), calculate the length of the side \( AC \).
\( \sqrt6\,\mathrm{cm} \)
\( 3\sqrt2\,\mathrm{cm} \)
\( 2\sqrt3\,\mathrm{cm} \)
\( 3\frac{\sqrt3}{\sqrt2}\,\mathrm{cm} \)

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.
\( 90^{\circ};\ 36.87^{\circ};\ 53.13^{\circ} \)
\( 90^{\circ};\ 30.96^{\circ};\ 59.04^{\circ} \)
\( 90^{\circ};\ 38.65^{\circ};\ 51.35^{\circ} \)
\( 90^{\circ};\ 33.13^{\circ};\ 56.87^{\circ} \)

1003077010

Level: 
B
In an isosceles triangle \( ABC \) the base \( AB \) has length \( 12\,\mathrm{cm} \). The altitude to the base \( v_c=8\,\mathrm{cm} \). Determine the length of the median drawn from a vertex at the base to the side.
\( \sqrt{97}\,\mathrm{cm} \)
\( \sqrt{93}\,\mathrm{cm} \)
\( \sqrt{87}\,\mathrm{cm} \)
\( \sqrt{83}\,\mathrm{cm} \)