Open the Geogebra applet Parallel/perpendicular lines. Draw a line k. Choose a way of constructing the line l when there are two points that this line goes through – the option ‘two points’. A line l going through two randomly chosen points will appear. By changing the location of any of these points establish the position of the line l in such a way that it is parallel to the line k. Using the applet, verify if it is actually parallel to the line k. What is the slopeslopeslope of the line l?

Open the geogebra applet parallel/perpendicular lines. Draw a line k. Choose a way of constructing the line l when there are two points that this line goes through – the option ‘two points’. A line l going through two randomly chosen points will appear. By changing the location of any of these points establish the position of the line l in such a way that it is perpendicular to the line k. Using the applet, verify if it is actually perpendicular to the line k. If the line l is perpendicular to the line k, then calculate the slope of the line l. Write the result in the table. Repeat this procedure a few times and write down results of following observations.

Open the geogebra applet parallel/perpendicular lines.  Draw a line k. Choose the way of constructing the line l when there is given slopeslopeslope of the line and a point that it goes through – the option ‘slope‑point’. After writing the value of the numerator and the denominator of the slope in the window of the applet, the line l occurs, that goes through the randomly chosen point. Using the slider set the value of the slope l in such a way that the line l is perpendicular to the line k. By changing the location of the point on the line l, you can move it parallely and set in the window of the applet in a convenient way. Using the applet verify if it actually perpendicular to the line k. If the line l is perpendicular to k, then write the set slope of the line l with the slope of the line k in the table. Repeat this procedure a few times and write down results of following observations.

There is the line k and point A. Find the equation of the line l going through the point A and parallel to k and equation of the line m going through point A and perpendicular to k.

a) $k:y=2x-1,A=(0,3)$

b) $k:y=-x+\frac{{\displaystyle 1}}{2},A=(0,-2)$

c) $k:y=\frac{{\displaystyle 2}}{3}x-2,A=(1,4)$

d) $k:y=-\frac{{\displaystyle 3}}{5}x+\frac{{\displaystyle 1}}{3},A=(-2,\frac{2}{5})$

There are lines k and l. Find all values m for which the line l is paralel to the line k and all values m for which the line l is perpendicular to the line k.

a) $k:y=5x,l:y=(2-m)x-1$

b) $k:y=-2x-\frac{{\displaystyle 3}}{2},l:y=\frac{{\displaystyle 1}}{2m}x-1$

c) $k:y=\frac{{\displaystyle 3}}{5}x-1,l:y=\frac{{\displaystyle m+1}}{m-1}x+m$

d) $k:y=-\frac{{\displaystyle 7}}{2}x-\frac{{\displaystyle 2}}{7},l:y=\frac{{\displaystyle 2m}}{3m+1}x+\frac{{\displaystyle m}}{m-3}$

Points $A=(-3,-2)$, $B=(5,2)$, $C=(2,4)$ and $D=(-2,2)$ are vertices of the teragon ABCD. Prove that this tetragon is a trapezium but not a right‑angled trapezium.

An extra task:

Prove that no line whose equation is $y=(m^2-1)x+\frac{{\displaystyle 1}}{2m^2-3}$ is either parallel or perpendicular to the line $y=-2x+1$.

Points $A=(-4,-3)$, $B=(4,1)$, $C=(8,9)$ and $D=(-1,6)$ are vertices of a tetragon ABCD. Prove that this tetragon is not a trapezium but its diagonals are perpendicular.

There is a line k whose equation is $k:y=\frac{2}{3}x+4$ and a point $A=(4,-1)$.

Identify the equation of the line l going through the point A and parallel to the line k and the equation of the line m going through the point A and perpendicular to the line k. Write down your line of reasoning in English.