Question

The diagonals of a parallelogram$PQRS$are along the lines$x+3y=4$ and $6x-2y=7$.Then$PQRS$ must be

• A. Rectangle
• B. Square
• C. Cyclic quadrilateral
• D. Rhombus

Answer 1

The correct option is D

Rhombus

Explanation for the correct answer:

Comparing slope of diagonals:

Given the equation of diagonals of a parallelogram are $x+3y=4$ and $6x-2y=7$ let us find the slopes of each line.

$x+3y=4$ comparing with $y=mx+c$ we get, $m=\frac{-1}{3}$

Finding slope of other diagonal $6x-2y=7$ by comparing with $y=m\text{'}x+c\text{'}$ we get, $m\text{'}=3$

So, we see that product of slope of each diagonal is $m\times m\text{'}=\frac{-1}{3}\times 3=-1$ which is the condition of perpendicularity of two lines.

So, both diagonals are perpendicular to each other which is a case of square, rectangle and rhombus, but nothing is said about diagonals are equal or not so the best we can say is that given parallelogram must be a rhombus as diagonals of a rhombus need not be equal.

Hence, the correct option is (D) Rhombus