It's your turn on geometry of the plane: points, segments, lines

Do your best in trying to solve the following problems. In any case some of them are solved in the video and all of them are solved in the pdf file below.

Exercise 1.

Find the equation of the line (r) passing through the point ((6,2)) which forms an angle (theta=dfrac{pi}6) with the (y)-axis. Compute the intersection (Q) of (r) with the (y)-axis and the equation of the line (s) passing through (Q) perpendicular to (r).

Exercise 2.

Given the points (A=(2,-3)), (B=(4,0)), (C=(-1,2)),

1. find the coordinates of the midpoint (M) of (AB) and of the midpoint (N) of (BC);

2. find the centroid (G) of (ABC).

Exercise 3.

Find the point of the form ((x,3/2)) that lies on the segment between ((-2,1)) and ((7,3)).

Exercise 4.

Find the values of (p) and (q) such that the lines [ r_p: pX-2Y-1=0 quadtext{and}quad s_q: 6X-4Y-q=0 ]

1. have exactly one point in common;

2. are parallel and distinct;

3. coincide.

Exercise 5.

In the cartesian plane consider the points (A(-2, -1)) and (B(1, 2)).

1. Find the equation of the line (r) passing through (A) and (B).

2. Find the point (C) of intersection between the line (r) and the (y)-axis.

3. Find the equation of the line (s) parallel to (r) and passing through the origin.