Question

LMNP is a parallelogram. what additional information would prove that LMNP is a Rectangle?

Answer 1

option D is the answer

hope it helps!!

Answer 2

(D) LP⊥PN

Explanation:

A rectangle is a parallelogram with four right angles., thus in order to prove that LMNP is rectangle, we have to show that LP is perpendicular to PN.

The coordinates of vertices are L(-4,1), P(-3,-1) and N(3,2), then

`PL=(-4+3,1+1)`

⇒
`PL=(-1,2)`

And,
`PN=(3+3,2+1)`

⇒
`PL=(6,3)`

Now, taking the dot product, we have

`PL{\cdot}PN=(-1)(6)+(2)(3)`

⇒
`PL{\cdot}PN=0`

Since the dot product of two vectors is equal to zero, these vectors are perpendicular.

Also, It is given that LMNP is a parallelogram , therefore

`m{\angle}P=m{\angle}M=90^(\circ)` and
`m{\angle}L=m{\angle}N=180^(\circ)-90^(\circ)=90^(\circ)`

Thus, all the angles of the given parallelogram are equal and are equal to 90°, therefore LMNP is a rectangle.

Hence proved.

Thus, option D is correct.