Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2009  OctNov  (P19709/12)  Q#9
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Question
The diagram shows a rectangle . The point
i. Explain why the ycoordinate of
ii. Express the gradients of
iii. Calculate the xcoordinates of
iv. Calculate the area of the rectangle
Solution
i.
Consider the diagram below.
We know that the two diagonals of the rectangle intersect at midpoint. Therefore, point E is the midpoint of both AC & BD, diagonals.
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
Therefore;
ycoordinate of midpoint
ycoordinate of midpoint
Since, E & D lay on same horizontal line i.e. BD parallel to xaxis. Their ycoordinates are same.
ii.
From (i), we have;
We are also given that
Therefore;
Expression for slope (gradient) of a line joining points
Therefore, slope/gradient of AD & AC;
iii.
From the given diagram, we can find the length of diagonal AC.
Expression to find distance between two given points
Therefore;
We know that diagonals of a rectangle are equal in length. Hence;
We have the ycoordinate of the midpoint of the diagonal BD, from (i);
ycoordinate of midpoint
We can also find xcoordinate of the midpoint of the diagonal BD;
xcoordinate of midpoint
xcoordinate of midpoint
Therefore;
Since;
We have 10 units length on each side of midpoint E on the diagonal BD. Therefore;
xcoordinate of D
xcoordinate of B
Therefore;
iv.
Expression for the area of the rectangle is;
For the given case;
Expression to find distance between two given points
Therefore;



Hence;
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