Question

The points A and B have coordinates (1,6) and (5,-2) respectively. The mid-point of AB is M.

a.   Find the coordinates of M.

b.   Find the gradient of AB, giving your answer in its simplest form.

c.   A straight line passes through M and is perpendicular to AB.

                    i.       Show that this line has equation  .

                  ii.       Given that this line passes through the point (k, k+5) , find the value of the constant k.

Solution

a.
 

We are given points A and B with coordinates (1,6) and (5,-2) respectively and required to find the  coordinates of their mid-point of M.

To find the mid-point of a line we must have the coordinates of the end-points of the line.

Expressions for coordinates of mid-point of a line joining points  and;

x-coordinate of mid-point  of the line

y-coordinate of mid-point  of the line

Therefore, for the given case;

x-coordinate of mid-point  of the line

y-coordinate of mid-point  of the line

Hence, coordinates of mid-point (M) of AB are (3,2).

b.    

We are given points A and B with coordinates (1,6) and (5,-2) respectively and required to find the  gradient of line AB.

Expression for slope (gradient) of a line joining points  and ;

Therefore, for the given case;

c.    

                            i.
 

We are given that a straight line, say , passes through M and is perpendicular to line AB.

We are required to write equation of line .

To find the equation of the line either we need coordinates of the two points on the line (Two-Point  form of Equation of Line) or coordinates of one point on the line and slope of the line (Point-Slope  form of Equation of Line).

From (a) we have coordinates of point M(3,2) but we need to find the slope of line .

We are given that line  is perpendicular to line AB.

If two lines are perpendicular (normal) to each other, then product of their slopes  and  is;

Therefore;

From (b) we have slope of the line AB as .

Hence;

Now with the coordinates of a point on line  as M(3,2) and slope of the line  as , we can  write equation of line .

Point-Slope form of the equation of the line is;

ii.
 

We are given that line  passes through a point (k,k+5).

If a line passes through a point, then coordinates of this point must satisfy the equation of the line.

From (c:i) we have found equation of line ;

Therefore, for point (k,k+5), equation must satisfy.