Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2014  OctNov  (P19709/12)  Q#9
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Question
The diagram shows a trapezium ABCD in which AB is parallel to DC and angle BAD is . The coordinates of A, B and C are , and respectively.
i. Find the equation of AD.
ii. Find, by calculation, the coordinates of D.
The point E is such that ABCE is a parallelogram.
iii. Find the length of BE.
Solution
i.
To find the equation of the line either we need coordinates of the two points on the line (TwoPoint form of Equation of Line) or coordinates of one point on the line and slope of the line (PointSlope form of Equation of Line).
For the given case, line we have the coordinates of a point on line
If two lines are perpendicular (normal) to each other, then product of their slopes
Hence;
To find the slope of
Expression for slope of a line joining points
For the given case;
Hence;
Now we can write the equation of the line
PointSlope form of the equation of the line is;
For the given case;
ii.
To calculate the coordinates of point
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
For the given case;
Equation of the line
Now we find the equation of
To find the equation of the line either we need coordinates of the two points on the line (TwoPoint form of Equation of Line) or coordinates of one point on the line and slope of the line (PointSlope form of Equation of Line).
For the given case, line
We have found slope of
Therefore;
Now we can write the equation of the line
PointSlope form of the equation of the line is;
For the given case;
Equation of the line
We can also manipulate this equation;
Equating both equations of lines
Single value of x indicates that there is only one intersection point.
With xcoordinate of point of intersection of two lines (or line and the curve) at hand, we can find the ycoordinate of the point of intersection of two lines (or line and the curve) by substituting value of xcoordinate of the point of intersection in any of the two equations.
We choose;
Hence coordinates of
iii.
We are given that ABCE is a parallelogram.
Consider the diagram below. If ABCE is a parallelogram, then BC and AE must be equal and parallel. We also know that diagonal of a parallelogram bisect each other.
Therefore, for the given case, AC and BE diagonals bisect at point
To find the midpoint of a line we must have the coordinates of the endpoints of the line.
Expressions for coordinates of midpoint of a line joining points
xcoordinate of midpoint
ycoordinate of midpoint
Since
xcoordinate of midpoint
ycoordinate of midpoint
Therefore,
Since












Therefore,
Now we can find length of BE.
Expression to find distance between two given points
For
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