1103076904
Level:
B
The picture shows an acute triangle \( ABC \). Its area \( S=\frac{\sqrt3|AB|\cdot|AC|}4 \). What is the measure of the angle \( \alpha \)?
\( 60^{\circ} \)
\( 45^{\circ} \)
\( 30^{\circ} \)
\( 90^{\circ} \)
Triangles
1103076907
Level:
B
\( ABC \) is a triangle with the sides of lengths \( c=15 \), \( b=6 \), and the measure of \( \measuredangle CAB \) is \( 150^{\circ} \). Which of the numbers gives the size of the angle \( BCA \) most accurately?
\( 21.55^{\circ} \)
\( 11.54^{\circ} \)
\( 5.77^{\circ} \)
\( 9.23^{\circ} \)
1103076908
Level:
B
The area of an obtuse triangle is \( 4\,\mathrm{cm}^2 \) and the lengths of the sides containing the obtuse angle are \( 2\,\mathrm{cm} \) and \( 8\,\mathrm{cm} \). Give the measure of this angle.
\( 150^{\circ} \)
\( 120^{\circ} \)
\( 135^{\circ} \)
\( 105^{\circ} \)
1103077003
Level:
B
In an isosceles triangle \( ABC \), the base \( AB = 6\,\mathrm{cm} \), the measure of the angle \( ABC \) is \( 20^{\circ} \). The angle bisector of the angle \( BAC \) intersects the side \( BC \) at a point \( K \). Find the length of the line segment \( BK \). Round the result to two decimal places.
\( 2.08\,\mathrm{cm} \)
\( 5.64\,\mathrm{cm} \)
\( 2.05\,\mathrm{cm} \)
\( 2.18\,\mathrm{cm} \)
1103077004
Level:
B
In a triangle \( ABC \), \( a=15\,\mathrm{cm} \), \( b=6\,\mathrm{cm} \) and the measure of the angle \( CAB \) is \( 120^{\circ} \). Which of the following numbers gives as accurately as possible the measure of the angle \( ABC \)?
\( 20.27^{\circ} \)
\( 25.66^{\circ} \)
\( 60^{\circ} \)
\( 30^{\circ} \)
1103077005
Level:
B
In a triangle \( ABC \), \( c=2\,\mathrm{cm} \), \( a=3\,\mathrm{cm} \) and \( \beta=60^{\circ} \). What is the length of side \( b \)?
\( \sqrt7\,\mathrm{cm} \)
\( 13-6\sqrt3\,\mathrm{cm} \)
\( 19\,\mathrm{cm} \)
\( 13+6\sqrt3\,\mathrm{cm} \)
1103077007
Level:
B
In a triangle \( ABC \), \( a:b=1:2 \) and the measure of the angle \( BAC \) is \( 30^{\circ} \). Determine the size of the smallest interior angle of the triangle.
\( 30^{\circ} \)
\( 20^{\circ} \)
\( 10^{\circ} \)
\( 90^{\circ} \)
1103077008
Level:
B
Given a triangle \( ABC \), the length of the median from \( C \) is \( 9\,\mathrm{cm} \) and the length of the median from \( B \) is \( 6\,\mathrm{cm} \). Let \( T \) be the centroid, and \( S \) be the midpoint of \( AC \). The measure of the angle \( BTC \) is \( 120^{\circ} \). Find the length of the side \( AC \).
\( 4\sqrt7\,\mathrm{cm} \)
\( 2\sqrt7\,\mathrm{cm} \)
\( 2\sqrt{13}\,\mathrm{cm} \)
\( 4\sqrt{13}\,\mathrm{cm} \)
1103077009
Level:
B
The angles \( \alpha \), \(\beta \), \( \gamma \) of a right-angled triangle \( ABC \) are in the ratio \( 1:2:3 \) (see the picture). From the following ratios of sides select the one that is equal to \( \sqrt3:1 \).
\( b:a \)
\( a:b \)
\( c:b \)
\( c:a \)
1103077011
Level:
B
Consider a triangle \( ABC \) with \( a=1\,\mathrm{cm} \) and \( b = \sqrt3\,\mathrm{cm} \). The angle opposite the longer side is double the angle opposite the shorter side. Find the area of the triangle.
\( \frac{\sqrt3}2\,\mathrm{cm}^2 \)
\( 2\sqrt3\,\mathrm{cm}^2 \)
\( \sqrt3\,\mathrm{cm}^2 \)
\( \frac{\sqrt3}4\,\mathrm{cm}^2 \)