question_answer

If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.

A)  x = 6, y = 8                

B)  x = 2, y = 3

C)  x = 6, y = 3    

D)  x =  5, y = 4

Correct Answer: C

Solution :

Let A (1, 2), B (4, y), C (x, 6) and D (3, 5) are the vertices of parallelogram.

Since, ABCD is a parallelogram.

\[\therefore \] AC and BD will bisect each other. Hence, mid- point of AC and mid-point of BD are same point.

\[\therefore \]Mid-point of AC is \[\left( \frac{1+x}{2},\frac{2+6}{2} \right)\]

= Mid-point of BD is\[\left( \frac{4+3}{2},\frac{y+5}{2} \right)\]

\[\left[ \because \text{Mid-point}=\left( \frac{{{x}_{1}}+{{x}_{2}}}{2},\frac{{{y}_{1}}+{{y}_{2}}}{2} \right) \right]\]

\[\therefore \]      \[\frac{1+x}{2}=\frac{4+3}{2}\]and\[\frac{2+6}{2}=\frac{5+y}{2}\]

\[\Rightarrow \]   \[1+x=7\]and\[8=5+y\]

\[x=6\]and\[y=3\]