\(A(-8,\,7)\), \(B(-3,\,-2)\) and \(C(9,\,1)\) are three vertices of a parallelogram ABCD, taken in order. What are the coordinates of D?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \((4,\,10)\)
The diagonals of a parallelogram bisect each other, so diagonals AC and BD have the same mid-point.
Mid-point of AC: \(\displaystyle \left(\frac{-8+9}{2},\,\frac{7+1}{2}\right)=\left(\frac{1}{2},\,4\right)\)
Let \(D=(x,\,y)\). Mid-point of BD: \(\displaystyle \left(\frac{-3+x}{2},\,\frac{-2+y}{2}\right)\)
\(-3+x=1\), so \(x=4\)
\(-2+y=8\), so \(y=10\)
So \(D=(4,\,10)\)
