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Question
Prove that the lines 2x − 3y + 1 = 0, x + y = 3, 2x − 3y = 2 and x + y = 4 form a parallelogram.
Solution
The given lines can be written as
\[y = \frac{2}{3}x + \frac{1}{3}\] ... (1)
\[y = - x + 3\] ... (2)
\[y = \frac{2}{3}x - \frac{2}{3}\] ... (3)
\[y = - x + 4\] ... (4)
The slope of lines (1) and (3) is \[\frac{2}{3}\] and that of lines (2) and (4) is −1.
Thus, lines (1) and (3), and (2) and (4) are two pair of parallel lines.
If both pair of opposite sides are parallel then ,we can say that it is a parallelogram.
Hence, the given lines form a parallelogram.
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