Analytical Plane Geometry

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

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.
\( \varphi\doteq 71^{\circ}34' \)
\( \varphi\doteq 71^{\circ}33' \)
\( \varphi\doteq 71^{\circ}40' \)
\( \varphi\doteq 71^{\circ}38' \)

9000090902

Level: 
C
Given the parametric line \(p\), find \(m\in \mathbb{R}\) such that the point \(C = [m,3]\) is on the line \(p\). \[ \begin{aligned}p\colon x& = 1 - t, & \\y & = -3 + 2t,\ t\in \mathbb{R} \\ \end{aligned} \]
\(m = -2\)
\(m = 4\)
\(m = 11\)
\(m = -\frac{11} {3} \)
\(m = \frac{3} {2}\)