Question

Figure 2 shows a right angled triangle LMN.

The points L and M have coordinates (–1, 2) and (7, –4) respectively.

a.   Find an equation for the straight line passing through the points L and M. Give your answer in the  form ax + by + c = 0, where a, b and c are integers.

Given that the coordinates of point N are (16, p), where p is a constant, and angle LMN = 90°,

b.   find the value of p.

Given that there is a point K such that the points L, M, N, and K form a rectangle, 

c.   find the y coordinate of K.

Solution

a)    

We are required to find equation of line passing through L(–1, 2) and M (7, –4).

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).

We have coordinates of both points L(–1, 2) and M (7, –4).

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

Therefore;

b)   

We are given that lines LM and MN are perpendicular.

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

Therefore, we need slopes of the two lines.

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

Hence;

We have coordinates of both points L(–1, 2), M (7, –4) and N(16,p).

c)    

We are given that there is a point K such that the points L, M, N, and K form a rectangle. 

We have coordinates of both points L(–1, 2), M (7, –4), N(16,8) and K(x,y).

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

Slope of line LM;

Slope of line MN;

Slope of line KN;

It is evident that line KN perpendicular to MN.

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

Therefore;

It is evident that line KN must be parallel to line LM.

If two lines are parallel to each other, then their slopes  and  are equal;

Therefore;

Now we have got following two equations and we can solve these to find y.

We can rearrange both equations for 3x.

We can equate both equations;