Analytical Plane Geometry

9000106805

Level:

Given points $A = [0;5]$, $B = [6;1]$, $C = [7;9]$, find the direction vector of the line passing through the point $A$ and the midpoint of the segment $BC$ (i.e. the median of the triangle $ABC$ through the vertex $A$).

$(1;0)$

$(1;8)$

$(1;9)$

$(6.5;5)$

9000107502

Level:

In the following list identify a line which is perpendicular to the line $q$. \[ \begin{aligned}q\colon x& = 5 - t,& \\y & = 3t;\ t\in \mathbb{R} \\ \end{aligned} \]

$p\colon x - 3y - 7 = 0$

$p\colon - x - 3y + 11 = 0$

$p\colon 3x - y = 0$

$p\colon y = 5$

9000107504

Level:

Find the angle between the lines \[ p\colon 2x - 3y + 1 = 0;\quad q\colon 3x + 2y - 3 = 0. \]

$90^{\circ }$

$60^{\circ }$

$0^{\circ }$

$30^{\circ }$

9000107506

Level:

Find $\cos \varphi $ where $\varphi $ is the angle between the lines $p$ and $q$. \[ p\colon y = 2x - 11;\quad q\colon y = \frac{1} {4}x \]

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

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

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

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

9000106203

Level:

In the following list identify a vector having the same direction as the parametric line $p$. \[ \begin{aligned} p\colon x & = -5, \\ y & = 5t;\ t\in \mathbb{R} \end{aligned}\]

$(0;1)$

$(5;5)$

$(5;0)$

$(-5;5)$

9000106204

Level:

In the following list identify a vector having the same direction as the parametric line $p$. \[ \begin{aligned} x & = 2t, \\ y & = 0;\ t\in \mathbb{R}. \\\end{aligned}\]

$(1;0)$

$(0;0)$

$(0;1)$

$(0;2)$

9000106207

Level:

In the following list identify a normal vector to the line $p$. \[ p\colon - x + 2y + 3 = 0 \]

$(1;-2)$

$(2;3)$

$(-1;3)$

$(2;-1)$

9000106208

Level:

In the following list identify a normal vector to the line $p$. \[ p\colon 2y - 1 = 0 \]

$(0;1)$

$(2;0)$

$(2;-1)$

$(2;1)$

9000106801

Level:

Find the vector in the direction of the following line. \[ 2x + 1 = 3y - 2 \]

$(3;2)$

$(-3;2)$

$(2;-3)$

$(-3;3)$

9000106209

Level:

In the following list find the vector having the same direction as the line $p$. \[ p\colon y = 2x + 1 \]

$(1;2)$

$(2;-1)$

$(2;1)$

$(1;-2)$

Analytical Plane Geometry

9000106805

9000107502

9000107504

9000107506

9000106203

9000106204

9000106207

9000106208

9000106801

9000106209

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