Plane geometry

1103109105

Level:

Let

p

and

q

be the lines with the equations

x - 2 y - 1 = 0

and

2 x + y - 12 = 0

respectively. Find all the points at the same distance of

\sqrt{5}

from

p

and

q

(see the picture).

[2; 3]

[6; 5]

[8; 1]

[4; - 1]

[2; 3]

[6; 5]

[8.5; 1]

[4.5; - 1]

[2; 3.5]

[6; 5.5]

[8; 1]

[4; - 1]

[2; 3]

[6; 5.5]

[8; 1.5]

[4; - 1]

1103109106

Level:

Find general form equations of all the lines passing through the point

M = [- 2; 4]

at the distance of

2

from the origin

O

(see the picture).

x + 2 = 0; 3 x + 4 y - 10 = 0

x - 2 = 0; 3 x + 4 y - 10 = 0

x + 2 = 0; 4 x - 3 y + 20 = 0

x - 2 = 0; 4 x - 3 y + 20 = 0

1103109107

Level:

Let

A B C

be a triangle (see the picture). Determine the angle

φ

between the height

v_{c}

and the median

t_{c}

. Give the angle rounded to minutes.

φ ≐ 21^{\circ} 48^{'}

φ ≐ 21^{\circ} 24^{'}

φ ≐ 21^{\circ} 36^{'}

φ ≐ 21^{\circ} 52^{'}

1103109108

Level:

Let

A B C

be a triangle (see the picture). Determine the angle

φ

between the height

v_{b}

and the angle bisector

o_{α}

. Give the angle rounded to minutes.

φ ≐ 71^{\circ} 34^{'}

φ ≐ 71^{\circ} 33^{'}

φ ≐ 71^{\circ} 40^{'}

φ ≐ 71^{\circ} 38^{'}

2010014207

Level:

Given two points

A = [2; 1]

and

B = [4; - 2]

, identify a number

m \in R

such that the point

C = [1; m]

is on the line

A B

m = \frac{5}{2}

m = - \frac{1}{2}

m = 2

m = \frac{1}{3}

2010014208

Level:

Given lines

p

and

q

, find

m \in R

such that the lines

p

and

q

are parallel.

p : x + 3 y + 4 = 0, q : m x - 2 y - 7 = 0

m = - \frac{2}{3}

m = 6

m = - \frac{1}{3}

m = \frac{1}{3}

9000090901

Level:

Given two points

A = [2; 5]

and

B = [- 3; 2]

, identify a number

m \in R

such that the point

C = [1; m]

is on the line

A B

m = \frac{22}{5}

m = 20

m = - 3

m = \frac{2}{3}

m = - \frac{5}{2}

9000090902

Level:

Given the parametric line

p

, find

m \in R

such that the point

C = [m; 3]

is on the line

p

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

m = - 2

m = 4

m = 11

m = - \frac{11}{3}

m = \frac{3}{2}

9000090903

Level:

Given the line

p : 3 x - 2 y + 11 = 0,

find

m \in R

such that the point

C = [m; 0]

is on the line

p

m = - \frac{11}{3}

m = - 1

m = 11

m = - \frac{1}{11}

m = 2

9000090904

Level:

Given lines

p

and

q

, find

m \in R

such that the lines

p

and

q

are parallel.

p : x - 2 y + 7 = 0, q : m x + 3 y - 11 = 0

m = - \frac{3}{2}

m = \frac{2}{3}

m = \frac{3}{2}

m = - \frac{2}{3}

another solution

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