Plane geometry

1003090804

Level: 
B
Find the distance between parallel lines \( p \) and \( q \) given by their parametric equations. \begin{align*} p\colon x&=3+3t, & q\colon x&=2-3s, \\ y&=-1+t;\ t\in\mathbb{R}; & y&=1-s;\ s\in\mathbb{R}. \end{align*}
\( \frac{7\sqrt{10}}{10} \)
\( \frac{\sqrt{10}}{2} \)
\( \frac{\sqrt{10}}{5} \)
\( \frac{5\sqrt{10}}{2} \)

1103061301

Level: 
B
Let \( ABC \) be a triangle (see the picture). Find the standard form equations of the lines \( t \), \( v \) and \( o \), where \( t \) contains the median to \( AB \), \( v \) contains the altitude to \( AB \) and \( o \) is the line of symmetry of \( AB \). Choose the option with all equations correct.
\( t\colon 2x+y-10=0 ;\ v\colon 4x+y-16=0;\ o\colon 4x+y-20=0 \)
\( t\colon 2x+y-10=0;\ v\colon x-4y+13=0;\ o\colon x-4y-5=0 \)
\( t\colon x-2y-5=0;\ v\colon 4x+y-16=0;\ o\colon 4x+y-20=0 \)
\( t\colon x-2y-5=0;\ v\colon x-4y+13=0;\ o\colon x-4y-5=0 \)

1103090801

Level: 
B
Find a general form equation of the straight line that passes through the point \( M=[2;3] \) and is parallel with the line of symmetry of the line segment \( AB \), where \( A=[-1;4] \), and \( B=\left[\frac52;-3\right] \) (see the picture).
\( x-2y+4=0 \)
\( 2x+y-7=0 \)
\( 3x+2y-12=0 \)
\( 2x-3y+5=0 \)

1103109001

Level: 
B
Let \( A \) be the point \( [4;3] \) and let the line \( p \) has the equation \( x-y+3=0 \). Find the coordinates of the point \( A' \) which is a mirror reflection of \( A \) about the line of symmetry \( p \) (see the picture).
\( A'=[0;7] \)
\( A'=[1;8] \)
\( A'=[-1;8] \)
\( A'=[-1;7] \)

1103109002

Level: 
B
Let \( A=[0;1] \), \( B=[4;-2] \) and \( S=[4;3] \) be the points (see the picture). Find the coordinates of the points \( C \) and \( D \) so that \( ABCD \) is a parallelogram with the centre \( S \).
\( C=[8;5]\text{, } D=[4;8] \)
\( C=[7;5]\text{, } D=[4;8] \)
\( C=[8;5]\text{, } D=[4;7] \)
\( C=[4;8]\text{, } D=[8;5] \)

1103109003

Level: 
B
Let \( 2x+6y-5=0 \) be the line \( p \) and \( x+3y-4=0 \) be the line \( o \), where \( p \) and \( o \) are parallel (see the picture). Find the general form equation of a line \( p' \) which is the reflection of the line \( p \) about the line of symmetry \( o \).
\( p'\colon 2x+6y-11=0 \)
\( p'\colon 2x+6y-2=0 \)
\( p'\colon 2x+6y+5=0 \)
\( p'\colon -2x-6y-11=0 \)

1103109004

Level: 
B
Let \( p \) be the line with the equation \( x-2y-1=0 \) and let \( S \) be the point with coordinates \( [2;2] \) (see the picture). Find the general form equation of the line \( p' \) which is the image of the line \( p \) in the point symmetry with the centre in \( S \).
\( p'\colon x-2y+5=0 \)
\( p'\colon 2x-4y+9=0 \)
\( p'\colon x-2y+4=0 \)
\( p'\colon x-2y+6=0 \)

1103109005

Level: 
B
Let \( p \) be the line with the equation \( x-2y+5=0 \) and let \( \vec{v} \) be the vector \( (3;-2) \) (see the picture). Find the general form equation of the line \( p' \) which is the image of the line \( p \) translated by the vector \( \vec{v} \).
\( p'\colon x-2y-2=0 \)
\( p'\colon 2x-4y-3=0 \)
\( p'\colon x-2y-1=0 \)
\( p'\colon 2x-4y+3=0 \)

1103109006

Level: 
B
Let \( p \) be the line with the equation \( x-2y-1=0 \). Find the general form equations of all lines parallel to \( p \) such that their distance from \( p \) equals to \( \sqrt5 \).
\( x-2y+4=0;\ x-2y-6=0 \)
\( x-2y+\sqrt5=0;\ x-2y-\sqrt5=0 \)
\( x-2y-1+\sqrt5=0;\ x-2y-1-\sqrt5=0 \)
\( x-2y+6=0;\ x-2y-4=0 \)