Triangles

1003076810

Level: 
A
Interior angles of a triangle \( ABC \) are in the ratio \( 2:3:4 \). A circle is inscribed into the triangle \( ABC \). Points of tangency divide the circle into three arcs. What is the ratio of the lengths of these arcs?
\( 5:6:7 \)
\( 4:5:6 \)
\( 2:3:4 \)
\( 3:4:5 \)

1103076809

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

1003076808

Level: 
B
In a triangle \( ABC \) the measure of \( \measuredangle CAB \) is \( 45^{\circ} \) and the measure of \( \measuredangle CBA \) is \( 60^{\circ} \). The altitude to side \( AB \) is \( 1\,\mathrm{cm} \) long. Calculate the area of the triangle \( ABC \) in \(\mathrm{cm}^2 \).
\( \frac{\sqrt3+1}{2\sqrt3} \)
\( \frac{\sqrt3+1}{\sqrt3} \)
\( \frac{\sqrt3+1}{2} \)
\( \frac{\sqrt3+1}{4} \)

1003076806

Level: 
A
Choose the false statement:
In a triangle the side opposite the smallest interior angle is the longest side of a triangle.
The sum of the interior angles of a triangle is \( 180^{\circ} \).
There is at most one obtuse interior angle in a triangle.
In a triangle the sum of any two sides is greater than the third side.

1003076801

Level: 
A
\( ABC \) is a triangle with sides \( a \), \( b \), \( c \). Let \( a\leq b\leq c \). Two of its interior angles have measures of \( 70^{\circ} \) and \( 50^{\circ} \). Which of the following statements about the triangle \( ABC \) is true?
The third interior angle is opposite the side \( b \).
The angle of the measure \( 70^{\circ} \) lies opposite the side \( a \).
The angle of the measure \( 50^{\circ} \) lies opposite the side \( b \).
The third interior angle is opposite the side \( c \).