Plane geometry

Parallel Lines

Submitted by ladislav.foltyn on Thu, 05/02/2019 - 14:03
Question: 
In the table, mark a cell, if the two corresponding lines are parallel to each other. \begin{align*} a\colon&\, \left\{\begin{array}{ll} x=3+t\text{, } & \\ y=-3-t; & t\in\mathbb{R}\end{array}\right. & b\colon&\, y=3x-2 & c\colon&\, 4x-2y+5=0 \\ d\colon&\, y=\frac23x-7 & e\colon&\, 2x+y-6=0 & f\colon&\, \left\{\begin{array}{ll} x=3+4t\text{, } & \\ y=\phantom{3\,}-6t; & t\in\mathbb{R}\end{array}\right. \end{align*}

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).
\([2;3] \), \([6;5] \), \([8;1] \), \([4;-1] \)
\([2;3] \), \([6;5] \), \([8.5;1] \), \([4.5;-1] \)
\([2;3.5] \), \([6;5.5] \), \([8;1] \), \([4;-1] \)
\([2;3] \), \([6;5.5] \), \([8;1.5] \), \([4;-1] \)