Straight Lines
Parallelogram — Sum of Coordinates
nta_pyq_2024_jan
Grade 11
Question:
Let $\alpha,\beta,\gamma,\delta\in\mathbb{Z}$ and let $A(\alpha,\beta)$, $B(1,0)$, $C(\gamma,\delta)$ and $D(1,2)$ be the vertices of a parallelogram $ABCD$. If $AB=\sqrt{10}$ and the points $A$ and $C$ lie on the line $3y=2x+1$, then $2(\alpha+\beta+\gamma+\delta)$ is equal to
Step-by-Step Solution
Key Concept: Diagonals of parallelogram bisect each other. Midpoint of $AC$ = midpoint of $BD$. Midpoint of $BD=((1+1)/2,(0+2)/2)=(1,1)$. So $\alpha+\gamma=2$ and $\beta+\delta=2$. $2(\alpha+\beta+\gamma+\delta)=2(2+2)=8$.
$2(\alpha+\beta+\gamma+\delta)=8$.
Correct Answer: 4