Triangles

1003021707

Level: 
A
Choose the false statement.
All altitudes in a right-angled triangle are perpendicular to one another.
The centroid of a triangle divides each median in the ratio \( 2:1 \).
The midline of a triangle is parallel to the third side and half as long.
The medians of a triangle intersect in a single point, the centroid of a triangle.

1103021706

Level: 
A
In the triangle \( ABC \), \( \alpha=80^{\circ} \) and \( \beta=70^{\circ} \) (see the picture). Determine the measure of the angle between the altitude to the side \( AB \) and the altitude to the side \( BC \).
\( 70^{\circ} \)
\( 120^{\circ} \)
\( 30^{\circ} \)
\( 60^{\circ} \)

1003021705

Level: 
A
Calculate the measures of interior angles \( \alpha \), \( \beta \) and \( \gamma \) of a triangle if \( \gamma=2\beta \) and \( \beta=3\alpha \).
\( \alpha=18^{\circ};\ \beta=54^{\circ};\ \gamma=108^{\circ} \)
\( \alpha=15^{\circ};\ \beta=45^{\circ};\ \gamma=90^{\circ} \)
\( \alpha=12^{\circ};\ \beta=54^{\circ};\ \gamma=111^{\circ} \)
\( \alpha=54^{\circ};\ \beta=18^{\circ};\ \gamma=108^{\circ} \)

1003021703

Level: 
A
The measure of an exterior angle of an isosceles triangle is \( 84^{\circ} \). Calculate the measures of all interior angles of the triangle.
\( 96^{\circ};\ 42^{\circ};\ 42^{\circ} \)
\( 84^{\circ};\ 48^{\circ};\ 48^{\circ} \)
\( 12^{\circ};\ 84^{\circ};\ 84^{\circ} \)
\( 96^{\circ};\ 96^{\circ};\ 12^{\circ} \)

1103021702

Level: 
A
Given the triangle \( ABC \) (see the picture), where \( \alpha:\beta=5:7 \) and the angle \( \gamma \) is by \( 42^{\circ} \) smaller than the angle \( \omega \), calculate the measure of \( \gamma \).
\( 108^{\circ} \)
\( 42^{\circ} \)
\( 30^{\circ} \)
\( 60^{\circ} \)

1003021701

Level: 
A
Interior angles of a triangle \( ABC \) are in the ratio \( \alpha:\beta:\gamma=2:4:6 \). Calculate the measures of these angles.
\( \alpha=30^{\circ};\ \beta=60^{\circ};\ \gamma=90^{\circ} \)
\( \alpha=20^{\circ};\ \beta=40^{\circ};\ \gamma=60^{\circ} \)
\( \alpha=15^{\circ};\ \beta=30^{\circ};\ \gamma=135^{\circ} \)
\( \alpha=90^{\circ};\ \beta=60^{\circ};\ \gamma=30^{\circ} \)

1103021412

Level: 
B
The figure shows a rectangular trapezium whose bases have lengths of \( 21\,\mathrm{cm} \) and \( 15\,\mathrm{cm} \), and the longer arm is \( 10\,\mathrm{cm} \) long. Calculate the sine of the smallest interior angle of the trapezium.
\( 0.8 \)
\( 0.6 \)
\( 53.13^{\circ} \)
\( 36.87^{\circ} \)

9000150504

Level: 
C
The object \(y\) is projected using a lens with foci at \(F\) and \(F'\). The focal length of the lens (the distance from the focus to the lens) \(f = 20\, \mathrm{cm}\). The distance from the object \(y\) to the lens \(a = 60\, \mathrm{cm}\). Find the distance from the lens to the image \(y'\).
\(30\, \mathrm{cm}\)
\(600\, \mathrm{cm}\)
\(\frac{20} {3} \, \mathrm{cm}\)
\(25\, \mathrm{cm}\)