2000005502
Level:
C
Which side of the quadrilateral shown in the picture is the longest?
\(a\)
\(b\)
\(c\)
\(d\)
Triangles
2010015206
Level:
C
The lengths of the sides in a triangle are \( a \), \( b \), \( c \) and the opposite angles are \( \alpha \), \( \beta \), \( \gamma \). Give the measure of \( \beta \) if \( b^2=a^2+c^2+ac\sqrt3 \).
\( 150^{\circ}\)
\( 30^{\circ}\)
\( 60^{\circ}\)
\( 120^{\circ}\)
2010015207
Level:
C
The lengths of the sides of a triangle are \( 4\,\mathrm{cm} \), \( 6\,\mathrm{cm} \) and \( 8\,\mathrm{cm} \). Calculate cosine of its greatest interior angle.
\( -\frac14\)
\( 104.47^{\circ}\)
\( 28.96^{\circ}\)
\(\frac78\)
2010015303
Level:
C
Triangle \( ABC \) is given (See the picture.). Choose the correct statement, if \(r\) is a radius of its circumcircle.
\( \frac{b}{\sin\beta} = 2r \)
\( \frac{a}{\sin \alpha}= \frac{\sin\beta}{b}\)
\( c \sin \alpha = b \sin \gamma \)
\( \frac{a}{\sin \alpha}= r\)
2010015304
Level:
C
Which of the following formulas applies to the given triangle \( ABC \)? (See the picture.)
\( c^2 = a^2+b^2 -2ab\cos\gamma \)
\( c^2 = a^2+b^2 +2ab\cos\gamma \)
\( c^2 = a^2+b^2 -2ab\cos\alpha \)
\( c^2 = a^2+b^2 -2ab\sin\beta \)
2010015305
Level:
C
In a triangle \( ABC \), \( a=15\,\mathrm{cm} \), \( c=8\,\mathrm{cm} \) 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 \( BCA \)?
\( 27.51^{\circ} \)
\( 16.12^{\circ} \)
\( 30.13^{\circ} \)
\( 12.45^{\circ} \)
2010015306
Level:
C
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 \( 1:\sqrt3 \).
\( a:b \)
\( b:a\)
\( c:b \)
\( a:c \)
9000036101
Level:
C
A \(3\, \mathrm{m}\) long
rod is in a slant position with respect to the observer's eye: one end is in the distance
\(20\, \mathrm{m}\) and the
other one \(18\, \mathrm{m}\).
Find the visual angle of the rod (the angle between the lines which connect the
observer's eye and the ends of the rod) and round to the nearest degrees.
\(7^{\circ }\)
\(3^{\circ }\)
\(45^{\circ }\)
\(83^{\circ }\)
9000036102
Level:
C
Three forces act on the same body in the same point and the total
force on the body is zero (the forces cancel). The first two forces are
\(8\, \mathrm{N}\) and
\(10\, \mathrm{N}\) and the angle between
these forces is \(55^{\circ }\).
Find the third force.
\(16\, \mathrm{N}\)
\(15\, \mathrm{N}\)
\(17\, \mathrm{N}\)
\(18\, \mathrm{N}\)
9000036103
Level:
C
Three forces \(F_{1}\),
\(F_{2}\) and
\(F_{3}\) act on the
same body in the same point and the total force on the body is zero (the forces cancel). The first
two forces are \(F_{1} = 8\, \mathrm{N}\)
and \(F_{2} = 10\, \mathrm{N}\) and the
angle between \(F_{1}\)
and \(F_{2}\) is
\(55^{\circ }\). Find the
angle between \(F_{3}\)
and \(F_{1}\).
Round your answer to the nearest degrees.
\(149^{\circ }\)
\(125^{\circ }\)
\(55^{\circ }\)
\(30^{\circ }\)