Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#9

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Question

The diagram shows a quadrilateral ABCD in which the point A is , the point B is  and the point C is . The diagonals AC and BD intersect at M. Angle   and . Calculate

    i.        the coordinates of M and D,

   ii.       the ratio .

Solution

    i.
 

First we find the coordinates of point M.

It is evident from the diagram that point M is the intersection of AC and BD.

If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e. coordinates of that point have same values on both lines (or on the line and the curve). Therefore, we can equate  coordinates of both lines i.e. equate equations of both the lines (or the line and the
curve).

So we can find the coordinates of point M if we have equations of AC and BD.

Now we find the equation of AC.

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 can use two point form of the equation of line because we have coordinates of two points on the line AC namely point A and point C.

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

We are given that the point A is , and the point C is .

Therefore;

Next we find the equation of BD.

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 can use point-slope form of equation of Line because we have coordinates of point B on the line BD, however, we need to find the slope of the line BD.

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

It is evident from the diagram that line BD is perpendicular to line AC.

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

For the given case;

Slope-Intercept form of the equation of the line;

Where  is the slope of the line.

We have the equation of line AC from which we can find the slope of this line.

Therefore,

Hence;

Now, using slope  and point B  we can write the equation of line BD.

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

For the given case;

Now that we have equations of both intersecting lines AC and BD, we can find the coordinates of their point of intersection.

Equating both equations;

Single value of x indicates that there is only one intersection point.

Corresponding values of y coordinate can be found by substituting values of x in any of the two equation i.e either equation of the line or equation of the curve.

We choose;

Hence .


ii.
 

To find the ratio between AM & MC lines we need to find length of both AM & MC.

Expression to find distance between two given points  and is:

For AM, we have the point A is  and the point M(5,2). Therefore;

For MC, we have the point M is  and the point C (9,4). Therefore;

Hence;

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