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Question
If P(2, 4), Q(0, 3), R(3, 6) and S(5, y) are the vertices of a parallelogram PQRS, then the value of y is
Options
7
5
-7
-8
Solution
It is given that P(2, 4), Q(0, 3), R(3, 6) and S(5, y) are the vertices of a parallelogram PQRS.
Join PR and QS, intersecting each other at O.
We know that the diagonals of the parallelogram bisect each other. So, O is the mid-point of PR and QS.
Coordinates of mid-point of PR = \[\left( \frac{2 + 3}{2}, \frac{4 + 6}{2} \right) = \left( \frac{5}{2}, \frac{10}{2} \right) = \left( \frac{5}{2}, 5 \right)\]
Coordinates of mid-point of QS = \[\left( \frac{0 + 5}{2}, \frac{3 + y}{2} \right) = \left( \frac{5}{2}, \frac{3 + y}{2} \right)\]
Now, these points coincides at the point O.
\[\therefore \left( \frac{5}{2}, \frac{3 + y}{2} \right) = \left( \frac{5}{2}, 5 \right)\]
\[ \Rightarrow \frac{3 + y}{2} = 5\]
\[ \Rightarrow 3 + y = 10\]
\[ \Rightarrow y = 7\]
Thus, the value of y is 7.
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