If A(1, 2), B(3, 4), C(5, 7) and D(x, y) are the vertices of a parallelogram ABCD, then the value of x + y is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 8
The diagonals of a parallelogram bisect each other, so the diagonals AC and BD have the same mid-point. In ABCD, A and C are opposite vertices, and so are B and D.
Mid-point of AC \(\displaystyle =\left(\frac{1+5}{2},\,\frac{2+7}{2}\right)=\left(3,\,\frac{9}{2}\right)\)
Mid-point of BD \(\displaystyle =\left(\frac{3+x}{2},\,\frac{4+y}{2}\right)\)
\(3+x=6\), so \(x=3\)
\(4+y=9\), so \(y=5\)
\(x+y=3+5=8\)
Shortcut: Since the mid-points are equal, A + C = B + D for the coordinates, so \(x+y=(1+2)+(5+7)-(3+4)=15-7=8\), without finding x and y.