Analytical plane geometry

1103109107

Level: 
C
Let \( ABC \) be a triangle (see the picture). Determine the angle \( \varphi \) between the height \( v_c \) and the median \( t_c \). Give the angle rounded to minutes.
\( \varphi\doteq 21^{\circ}48' \)
\( \varphi\doteq 21^{\circ}24' \)
\( \varphi\doteq 21^{\circ}36' \)
\( \varphi\doteq 21^{\circ}52' \)

1103109108

Level: 
C
Let \( ABC \) be a triangle (see the picture). Determine the angle \( \varphi \) between the height \( v_b \) and the angle bisector \( o_\alpha \). Give the angle rounded to minutes.
\( \varphi\doteq 71^{\circ}34' \)
\( \varphi\doteq 71^{\circ}33' \)
\( \varphi\doteq 71^{\circ}40' \)
\( \varphi\doteq 71^{\circ}38' \)

1103109105

Level: 
C
Let \( p \) and \( q \) be the lines with the equations \( x-2y-1=0 \) and \( 2x+y-12=0 \) respectively. Find all the points at the same distance of \( \sqrt5 \) from \( p \) and \( q \) (see the picture).
\([2;3] \), \([6;5] \), \([8;1] \), \([4;-1] \)
\([2;3] \), \([6;5] \), \([8.5;1] \), \([4.5;-1] \)
\([2;3.5] \), \([6;5.5] \), \([8;1] \), \([4;-1] \)
\([2;3] \), \([6;5.5] \), \([8;1.5] \), \([4;-1] \)

1103109104

Level: 
C
Let \( 2x-3y+6=0 \) be the equation of the line \( p \) and let \( M \) be the point \( [5;3] \). Find equations of all lines passing through \( M \) and intersecting \( p \) at an angle of \( 45^{\circ} \) (see the picture).
\( x+5y-20=0;\ 5x-y-22=0 \)
\( x+6y-23=0;\ 6x-y-27=0 \)
\( x+4y-17=0;\ 4x-y-16=0 \)
\( x+5y-28=0;\ 5x-y-10=0 \)

1103109103

Level: 
C
Let \( y=-\frac{\sqrt3}3x+1 \) be the equation of the line \( p \) and let \( M \) be the point \( [0;-3] \). Find equations of all lines passing through \( M \) and intersecting \( p \) at an angle of \( 60^{\circ} \) (see the picture).
\( x=0;\ y=\frac{\sqrt3}3x-3 \)
\( y=0;\ y=\frac{\sqrt3}3x-3 \)
\( y=0;\ y=x-3 \)
\( x=0;\ y=\sqrt3x-3 \)

1103109102

Level: 
C
Let \( p \) and \( q \) be intersecting lines with the equations \( y=\frac{\sqrt3}3x \) and \( x=0 \) respectively. Find equations of lines \( o_1 \) and \( o_2 \) that are lines of symmetry of the angles contained between \( p \) and \( q \) (see the picture).
\( y=\sqrt3x;\ y=-\frac{\sqrt3}3x \)
\( y=2x;\ y=-\frac12x \)
\( y=\sqrt2x;\ y=-\frac{\sqrt2}2x \)
\( y=3x;\ y=-\frac13x \)

1103109101

Level: 
C
Find equations of all lines at the distance \( \sqrt{10} \) from the point \( M=[5;4] \) which are perpendicular to the line \( p \) with the equation \( 2x+6y-3=0 \) (see the picture).
\( 3x-y-1=0;\ 3x-y-21=0 \)
\( 3x-y+1=0;\ 3x-y-18=0 \)
\( x+3y+1=0;\ x+3y+21=0 \)
\( x+3y-1=0;\ x+3y-18=0 \)

1103061207

Level: 
A
We are given the straight line \( m= \left\{[3-t;t]\text{, } t\in\mathbb{R} \right\} \) which intersects lines \( a \), \( b \), \( c \) in points \( A \), \( B \), \( C \) consecutively (see the picture). Find the values of a parameter \( t \) corresponding to these line intersections.
\( t_A=1; t_B=\frac32;\ t_C=2 \)
\( t_A=-1; t_B=-2;\ t_C=-3 \)
\( t_A=2; t_B=\frac32;\ t_C=1 \)
\( t_A=2; t_B=\frac52;\ t_C=3 \)