Triangles

1003076906

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 \( \alpha \) if \( a^2 = b^2 + c^2 +bc \).
\( 120^{\circ} \)
\( 60^{\circ} \)
\( 30^{\circ} \)
\( 90^{\circ} \)

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 \( 140^{\circ} \).
\( 70^{\circ} \)
\( 60^{\circ} \)
\( 50^{\circ} \)
\( 40^{\circ} \)

1003076710

Level: 
C
\( ABC \) is a triangle where the side \( b \) is \( 74\,\mathrm{cm} \) long and the angle \( \alpha = 60^{\circ} \). Find the length of its side \( c \) if you know that the area of the triangle is \( 720.9\,\mathrm{cm}^2 \).
\( 22.5\,\mathrm{cm} \)
\( 37.56\,\mathrm{cm} \)
\( 38.97\,\mathrm{cm} \)
\( 24.54\,\mathrm{cm} \)

1003076708

Level: 
C
The measures of the interior angles of a triangle are \( 30^{\circ} \), \( 45^{\circ} \) and \( 105^{\circ} \). The length of its longest side is \( 10\,\mathrm{cm} \). The length of its shortest side is:
\( 5.18\,\mathrm{cm} \)
\( 7.33\,\mathrm{cm} \)
\( 5.01\,\mathrm{cm} \)
\( 7.07\,\mathrm{cm} \)

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.
\( 1.64\,\mathrm{cm} \)
\( 0.82\,\mathrm{cm} \)
\( 0.20\,\mathrm{cm} \)
\( 0.12\,\mathrm{cm} \)

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.
\( 1.86\,\mathrm{cm} \)
\( 2.14\,\mathrm{cm} \)
\( 3.12\,\mathrm{cm} \)
\( 4.61\,\mathrm{cm} \)