Triangles

1103021906

Level: 
C
The distance between the places \( A \) and \( C \) on a straight road is \( 300\,\mathrm{m} \). There is a balloon \( B \) above the road between the places \( A \) and \( C \). See the picture. The angles of elevation from the places \( A \) and \( C \) to the balloon \( B \) are \( 20^{\circ} \) and \( 40^{\circ} \) respectively. What is the height \( h \) of the balloon?
\( 76\,\mathrm{m} \)
\( 168\,\mathrm{m} \)
\( 488\,\mathrm{m} \)
\( 523\,\mathrm{m} \)

1103021907

Level: 
C
A plane is flying at a speed of \( 900\,\mathrm{km}\cdot\mathrm{h}^{-1} \) and according to the compass the axis of the airplane is still heading westbound. What angle will the flight path of the airplane make with the east-west direction if the south wind starts to blow at a speed of \( 10\,\mathrm{m}\cdot\mathrm{s}^{-1} \)? Round the result to two decimal places.
\( 2.29^{\circ} \)
\( 0.64^{\circ} \)
\( 0.01^{\circ} \)
\( 87.71^{\circ} \)

1103076809

Level: 
C
The diagram shows a square inscribed into an equilateral triangle with the side \( 4\,\mathrm{cm} \) long. Calculate the length of the side of the square. Round the result to two decimal places.
\( 1.86\,\mathrm{cm} \)
\( 2.14\,\mathrm{cm} \)
\( 3.12\,\mathrm{cm} \)
\( 4.61\,\mathrm{cm} \)

1103076811

Level: 
C
A circle is inscribed into an isosceles triangle. The base of the triangle is \( 4\,\mathrm{cm} \) long and the length of the altitude to the base is \( 10\,\mathrm{cm} \). Calculate the radius of the circle.
\( 1.64\,\mathrm{cm} \)
\( 0.82\,\mathrm{cm} \)
\( 0.20\,\mathrm{cm} \)
\( 0.12\,\mathrm{cm} \)

1103076905

Level: 
C
The triangle in the diagram is divided into two isosceles triangles \( AKC \) and \( KBC \) which have the same area. Calculate the size of the angle \( \beta \) if you know that the measure of \(\measuredangle AKC \) is \( 140^{\circ} \).
\( 70^{\circ} \)
\( 60^{\circ} \)
\( 50^{\circ} \)
\( 40^{\circ} \)

1103076907

Level: 
C
\( ABC \) 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: 
C
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.
\( 150^{\circ} \)
\( 120^{\circ} \)
\( 135^{\circ} \)
\( 105^{\circ} \)