Plane geometry

2010014610

Level:

Given points

A = [4; - 1]

B = [2, - 3]

and

C = [5, 5]

, find the angle

β

(the interior angle at the vertex

B

) in the triangle

A B C

24^{\circ} 27^{'}

144^{\circ} 46^{'}

155^{\circ} 33^{'}

11^{\circ} 05^{'}

9000107504

Level:

Find the angle between the lines

p : 2 x - 3 y + 1 = 0; q : 3 x + 2 y - 3 = 0.

90^{\circ}

60^{\circ}

0^{\circ}

30^{\circ}

9000107505

Level:

Find

\cos φ

where

φ

is the angle between the lines

p

and

q

\begin{aligned} p : x & = 1 + 4 t, \\ y & = 3 - 3 t; t \in R; \end{aligned} q : x + y - 3 = 0

\frac{7 \sqrt{2}}{10}

- \frac{7}{5 \sqrt{2}}

\frac{\sqrt{2}}{5}

\frac{\sqrt{2}}{10}

9000107506

Level:

Find

\cos φ

where

φ

is the angle between the lines

p

and

q

p : y = 2 x - 11; q : y = \frac{1}{4} x

\frac{6 \sqrt{85}}{85}

\frac{1}{\sqrt{22}}

\frac{\sqrt{6}}{85}

\frac{\sqrt{17}}{30}

9000107507

Level:

Find

tg φ

where

φ

is the angle between the lines

p

and

q

\begin{aligned} p : x & = 1 + t, \\ y & = 3 + 2 t; t \in R; \end{aligned} q : y = 1

2

\frac{1}{2}

- 1

0

9000107508

Level:

Find

\cos φ

where

φ

is the angle between the lines

p

and

q

\begin{aligned} p : x & = t, \\ y & = - 3; t \in R; \end{aligned} q : y = 1

1

\frac{1}{\sqrt{2}}

0

\frac{\sqrt{10}}{10}

9000107509

Level:

In the following list identify a parametric line such that the angle between this line and the line

q

0^{\circ}

q : x - 2 y + 11 = 0

\begin{aligned} p : x & = 1 + 4 t, \\ y & = 3 + 2 t; t \in R \end{aligned}

\begin{aligned} p : x & = 1 + 2 t, \\ y & = 2 - t; t \in R \end{aligned}

\begin{aligned} p : x & = 2 - t, \\ y & = 3 + 2 t; t \in R \end{aligned}

\begin{aligned} p : x & = t, \\ y & = 1 - 2 t; t \in R \end{aligned}

9000149401

Level:

Find the distance from the point

P = [- 4; 2]

to the line

3 x - 4 y - 5 = 0

5

1

The line contains

P

\sqrt{5}

9000149402

Level:

Find the distance from the origin (i.e. the point

[0.0]

) to the line

p : x + 2 y + 5 = 0

\sqrt{5}

1

The origin is on the line

p

8

9000149403

Level:

Find the distance from the point

M = [1; 1]

to the line

p

\begin{aligned} p : x & = 3 + t, \\ y & = 1 + t; t \in R \end{aligned}

\sqrt{2}

2

1

0

(the point is on the line

p

)

2010014610

9000107504

9000107505

9000107506

9000107507

9000107508

9000107509

9000149401

9000149402

9000149403

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