Question

Prove that the points $(-2,-1),(1,0),(4,3)and(1,2)$ are the vertices of a parallelo-gram. Is it a rectangle?

Answer 1

Let the given point be A,B,C and D respectively. Then,

Coordinates of the mid-point of AC are $(\frac{-2+4}{2},\frac{-1+3}{2})=(1,1)$

Coordinates of the mid-point of BD are$\left({\frac{1+1}{2}}^{\prime }\frac{0+2}{2}\right)=(1,1)$

Thus, AC and BD have the same mid-point. Hence, ABCD is a parallelogram. Now, we shall see whether ABCD is a rectangle or not.

We have

,$AC=\sqrt{{(4-(-2))}^2+{(3-(-1))}^2}=2\sqrt{13}$

$and,BD=\sqrt{{(1-1)}^2+{(0-2)}^2}=2$
Clearly, $AC\ne BD.$ So, ABCD is not a rectangle.