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Question
Prove that the points (a, b), (a + 3, b + 4), (a − 1, b + 7) and (a − 4, b + 3) are the vertices of a parallelogram.
Solution
PQ = `sqrt (("a" + 3 - "a")^2 + ("b" + 4 - "b")^2) = sqrt (9 + 16) = 5` units
QR = `sqrt (("a" + 3 - "a" + 1)^2 + ("b" - 4 - "b" - 7)^2) = sqrt (16 + 9) = 5` units
RS = `sqrt (("a" -1 - "a" + 4)^2 + ("b" + 7 - "b" - 3)^2) = sqrt (9 + 16) = 5` units
SP = `sqrt (("a" - 4 - "a")^2 + ("b" + 3 - "b")^2) = sqrt (16 + 9) = 5` units
Since the opposite sides of quadrilateral PQRS are equal, therefore it is a parallelogram.
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