9000121704
Level:
A
Find the measure of the angle \(CAB\)
in an isosceles triangle \(ABC\)
with the interior angle \( BCA\)
of the measure \(120^{\circ }\).
\(30^{\circ }\)
\(60^{\circ }\)
\(15^{\circ }\)
\(45^{\circ }\)
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 \( BCX\).
\(50^{\circ }\)
\(80^{\circ }\)
\(100^{\circ }\)
\(40^{\circ }\)
9000045702
Level:
B
Given a right triangle \(ABC\) (see the picture). Find the
valid relation between the angle \(\alpha \)
and the sides of the triangle.
\(\mathop{\mathrm{tg}}\nolimits \alpha = \frac{a}
{c}\)
\(\sin \alpha = \frac{a}
{c}\)
\(\cos \alpha = \frac{b}
{a}\)
\(\mathop{\mathrm{cotg}}\nolimits \alpha = \frac{b}
{a}\)
9000045703
Level:
B
Given a right triangle \(ABC\) with the right angle at $C$ and an altitude $v$ (see the picture). Find the
valid relation between the angle \(\alpha \)
and the lengths in the triangle.
\(\sin \alpha = \frac{v}
{b}\)
\(\sin \alpha = \frac{v}
{c}\)
\(\sin \alpha = \frac{a}
{v}\)
\(\sin \alpha = \frac{c}
{a}\)
9000045704
Level:
B
Given a right triangle \(ABC\) with the right angle at $C$ and an altitude $v$ (see the picture). Find the
valid relation between the angle \(\beta \)
and the lengths in the triangle.
\(\sin \beta = \frac{v}
{a}\)
\(\mathop{\mathrm{tg}}\nolimits \beta = \frac{a}
{v}\)
\(\cos \beta = \frac{v}
{a}\)
\(\mathop{\mathrm{tg}}\nolimits \beta = \frac{v}
{a}\)
9000046403
Level:
B
Consider an isosceles triangle, i.e. the triangle with two
sides of equal length. The length of the third side is
\(4\, \mathrm{cm}\). One of the
interior angles is \(120^{\circ }\).
Find the area of this triangle.
\(\frac{4\sqrt{3}}
{3} \, \mathrm{cm}^{2}\)
\(4\sqrt{3}\, \mathrm{cm}^{2}\)
\(\frac{8\sqrt{3}}
{3} \, \mathrm{cm}^{2}\)
9000045701
Level:
B
The right angled triangle \(ABC\) is given (see the picture). Choose the correct statement.
\(\cos \beta = \frac{a}
{c}\)
\(\cos \beta = \frac{b}
{c}\)
\(\mathop{\mathrm{tg}}\nolimits \alpha = \frac{b}
{a}\)
\(\sin \alpha = \frac{c}
{a}\)
9000046401
Level:
B
Consider the rectangle \(ABCD\)
with length and height \(|AB| = 6\, \mathrm{cm}\)
and \(|BC| = 2\sqrt{3}\, \mathrm{cm}\), respectively.
Find the measure of the angle \( CAB\).
\(30^{\circ }\)
\(45^{\circ }\)
\(60^{\circ }\)
9000046402
Level:
B
Consider the rectangle \(ABCD\)
with length and height \(|AB| = 6\, \mathrm{cm}\)
and \(|BC| = 2\sqrt{3}\, \mathrm{cm}\). Let
\(S\) be the intersection of
diagonals. Find the angle \(\measuredangle ASB\).
\(120^{\circ }\)
\(60^{\circ }\)
\(90^{\circ }\)
9000045705
Level:
A
Find the formula for the radius \(r\)
of the circle circumscribed to the right triangle.
\(r = \frac{a}
{2\cdot \sin \alpha } \)
\(r = \frac{2a}
{\sin \alpha } \)
\(r = \frac{a}
{\sin 2\alpha } \)
\(r = \frac{a}
{\sin \alpha } \)