Triangles

2010012810

Level: 
A
In the triangle \( KLM \), \( k=10\,\mathrm{cm} \), \( l=8\,\mathrm{cm} \), \( m=12\,\mathrm{cm} \). Point \( N \) is the foot of the altitude from the vertex \( K \) (See the picture.) What is the radius of the circumcircle of the triangle \( KLN \)?
\( 6\,\mathrm{cm} \)
\( 5\,\mathrm{cm} \)
\( 7\,\mathrm{cm} \)
\( 8\,\mathrm{cm} \)

2000005604

Level: 
B
A right-angled triangle \(ABC\) with the right angle at the vertex \(C\) is given in the picture. Calculate the height \(v_c\), if \(a=20\,\mathrm{cm}\) and \(\beta=50^{\circ}\).
\( 20\sin{50^{\circ}}\)
\( 20\cos{50^{\circ}}\)
\( 20 \mathop{\mathrm{tg}} {50^{\circ}}\)
\( \frac{20}{\sin{50^{\circ}}}\)

2000005603

Level: 
B
A right-angled triangle \(ABC\) with the right angle at the vertex \(C\) is given in the picture. Calculate the length of the side \(b\), if \(a=20\,\mathrm{cm}\) and \(\beta=34^{\circ}\).
\(b=\frac{20}{ \mathop{\mathrm{tg}} {56^{\circ}}}\,\mathrm{cm}\)
\(b=\frac{20}{ \mathop{\mathrm{tg}} {34^{\circ}}}\,\mathrm{cm}\)
\( b=20\sin{34^{\circ}}\,\mathrm{cm}\)
\( b=20\cos{34^{\circ}}\,\mathrm{cm}\)

2000005602

Level: 
B
A right-angled triangle \(ABC\) is given in the picture. Its hypotenuse is \(10\,\mathrm{cm}\) long and the measure of its internal angle \(\alpha\) is \(30^{\circ}\). What are the lengths of the legs in the triangle?
\( a=5\,\mathrm{cm}\), \( b=5\sqrt{3}\,\mathrm{cm}\)
\( a=5\sqrt{3}\,\mathrm{cm}\), \( b=5\,\mathrm{cm}\)
\(a=10\cos{30^{\circ}}\,\mathrm{cm}\), \(b=10\sin{35^{\circ}}\,\mathrm{cm}\)
\(a=\frac{\sin{30^{\circ}}}{10}\,\mathrm{cm}\), \(b=\frac{\cos{30^{\circ}}}{10}\,\mathrm{cm}\)

2000005601

Level: 
B
A right-angled triangle \(ABC\) with the right angle at the vertex \(C\) is given in the picture. The length of the side \(c\) is \(5\,\mathrm{cm}\) and the measure of the angle \(\alpha\) is \(35^{\circ}\). What is the length of the side \(a\)?
\(5\sin{35^{\circ}}\,\mathrm{cm}\)
\(\frac{5}{\sin{35^{\circ}}}\,\mathrm{cm}\)
\(5\cos{35^{\circ}}\,\mathrm{cm}\)
\(\frac{5}{\cos{35^{\circ}}}\,\mathrm{cm}\)

2000003202

Level: 
A
An isosceles triangle \(ABC\) is given in the picture. What are the measures of its interior angles?
\( \alpha=27^{\circ};~\beta=27^{\circ};~\gamma=126^{\circ}\)
\( \alpha=54^{\circ};~\beta=54^{\circ};~\gamma=72^{\circ}\)
\( \alpha=63^{\circ};~\beta=63^{\circ};~\gamma=153^{\circ}\)
\( \alpha=126^{\circ};~\beta=27^{\circ};~\gamma=27^{\circ}\)