Triangles

2010015303

Level:

Triangle

A B C

is given (See the picture.). Choose the correct statement, if

r

is a radius of its circumcircle.

\frac{b}{\sin β} = 2 r

\frac{a}{\sin α} = \frac{\sin β}{b}

c \sin α = b \sin γ

\frac{a}{\sin α} = r

2010015304

Level:

Which of the following formulas applies to the given triangle

A B C

? (See the picture.)

c^{2} = a^{2} + b^{2} - 2 a b \cos γ

c^{2} = a^{2} + b^{2} + 2 a b \cos γ

c^{2} = a^{2} + b^{2} - 2 a b \cos α

c^{2} = a^{2} + b^{2} - 2 a b \sin β

2010015305

Level:

In a triangle

A B C

a = 15 cm

c = 8 cm

and the measure of the angle

C A B

120^{\circ}

. Which of the following numbers gives as accurately as possible the measure of the angle

B C A

{27.51}^{\circ}

{16.12}^{\circ}

{30.13}^{\circ}

{12.45}^{\circ}

2010015306

Level:

The angles

α

β

γ

of a right-angled triangle

A B C

are in the ratio

1 : 2 : 3

(See the picture.). From the following ratios of sides select the one that is equal to

1 : \sqrt{3}

a : b

b : a

c : b

a : c

2010015307

Level:

The angle of elevation of a straight road is

9^{\circ}

. The distance between two places measured along the road is

2 km

. For these places, find the difference in altitudes, i.e. the vertical distance, and round the result to the nearest meter. (See the picture.)

313 m

1975 m

317 m

78 m

2010015308

Level:

A roof gable has the shape of an isosceles triangle with the base of

14 m

and the height

6 m

. What is the angle between the roof and the horizontal direction? (See the picture.) Round your result to one decimal place.

{40.6}^{\circ}

{49.4}^{\circ}

{59.0}^{\circ}

{31.0}^{\circ}

2010015309

Level:

The right angled triangle

A B C

is given (See the picture.). Choose the correct statement.

tg α = \frac{a}{b}

\sin β = \frac{a}{c}

\cos β = \frac{a}{b}

tg β = \frac{a}{b}

9000035001

Level:

The angle of elevation of a straight road is

3^{\circ} 30^{'}

. The distance between two places measured along the road is

2 km

. For these places, find the difference in altitudes, i.e. the vertical distance, and round the result to the nearest meter. (See the picture.)

122 m

276 m

98 m

49 m

9000035003

Level:

The tree of the height

12 m

is observed from the place horizontal with the base of the tree. The angle of elevation is

10^{\circ}

. Find the distance of the observer from the base and round to the nearest meters.

68 m

2 m

12 m

48 m

9000035004

Level:

The triangle

A B C

has the angle

β = 59^{\circ}

and the side

a = 14 cm

. Find the altitude

v_{c}

(the line segment which is perpendicular to the side

c

and joins the vertex

C

with the side

c

) and round to the nearest centimeters.

12 cm

7 cm

10 cm

23 cm

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