Prove that the points (−4,−1), (−2, 4),

Question:

Prove that the points (−4,−1), (−2, 4), (4, 0) and (2, 3) are the vertices of a rectangle.

Solution:

Let A (−4,−1); B (−2,−4); C (4, 0) and D (2, 3) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a rectangle.

So we should find the lengths of opposite sides of quadrilateral ABCD.

$\mathrm{AB}=\sqrt{(-2+4)^{2}+(-4+1)^{2}}$

$=\sqrt{4+9}$

$=\sqrt{13}$

$\mathrm{CD}=\sqrt{(4-2)^{2}+(0-3)^{2}}$

$=\sqrt{4+9}$

$=\sqrt{13}$

Opposite sides are equal. So now we will check the lengths of the diagonals.

$\mathrm{AC}=\sqrt{(4+4)^{2}+(0+1)^{2}}$

$=\sqrt{64+1}$

$=\sqrt{65}$

$\mathrm{BD}=\sqrt{(2+2)^{2}+(3+4)^{2}}$

$=\sqrt{16+49}$

$=\sqrt{65}$

Opposite sides are equal as well as the diagonals are equal. Hence ABCD is a rectangle.