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Question
Without using the distance formula, show that the points A(4, 5), B(1, 2), C(4, 3) and D(7, 6) are the vertices of a parallelogram.
Solution
The given points are A(4, 5), B(1, 2), C(4, 3) and D(7, 6).
Slope of AB = `(2 - 5)/(1 - 4)`
Slope of AB = `(-3)/(-3)`
Slope of AB = 1
Slope of CD = `(6 - 3)/(7 - 4)`
Slope of CD = `3/3`
Slope of CD = 1
Since, slope of AB = slope of CD
Therefore, AB || CD
Slope of BC = `(3 - 2)/(4 - 1) = 1/3`
Slope of DA = `(5 - 6)/(4 - 7) = (-1)/(-3) = 1/3`
Since, slope of BC = slope of DA
Therefore, BC || DA
Hence, ABCD is a parallelogram
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