Triangles

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.
\(\frac{4\sqrt{3}} {3} \, \mathrm{cm}^{2}\)
\(4\sqrt{3}\, \mathrm{cm}^{2}\)
\(\frac{8\sqrt{3}} {3} \, \mathrm{cm}^{2}\)

9000150501

Level: 
B
A man of height \(180\, \mathrm{cm}\) casts a \(200\, \mathrm{cm}\) shadow. At the same moment, a tree of an unknown height casts a \(35\, \mathrm{m}\) shadow. Find the height of the tree.
\(\frac{63} {2} \, \mathrm{m}\)
\(\frac{350} {9} \, \mathrm{m}\)
\(\frac{72} {7} \, \mathrm{m}\)
\(\frac{36} {35}\, \mathrm{m}\)

1003021805

Level: 
C
In a triangle with interior angles \( 30^{\circ} \), \( 60^{\circ} \) and \( 90^{\circ} \), the longest side is \( 10\,\mathrm{cm} \) long. Find the length of its shortest side.
\( 5\,\mathrm{cm} \)
\( 5\sqrt3\,\mathrm{cm} \)
\( 3\,\mathrm{cm} \)
\( 8\,\mathrm{cm} \)

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:
\( 5.18\,\mathrm{cm} \)
\( 7.33\,\mathrm{cm} \)
\( 5.01\,\mathrm{cm} \)
\( 7.07\,\mathrm{cm} \)

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 \( 720.9\,\mathrm{cm}^2 \).
\( 22.5\,\mathrm{cm} \)
\( 37.56\,\mathrm{cm} \)
\( 38.97\,\mathrm{cm} \)
\( 24.54\,\mathrm{cm} \)