Question

Quadrilateral RSTU has vertices R(1,3), S(4,1), T(1,-3) and U(-2,-1). Is it true or false that this is a rectangle because the diagonals are congruent and the sides RS and ST are perpendicular?

Answer 1

Given vertices ofQuadrilateral RSTU as R(1,3), S(4,1), T(1,-3) and U(-2,-1).

We need to check if diagonals are congruent.

The coordinates of verticales diagonal RT are R(1,3) and  T(1,-3).

The coordinates of verticales diagonal SU are S(4,1), and  U(-2,-1),

By applying distance formula:

$\mathrm{Compute\:the\:distance\:between\:}\left(x_1,\:y_1\right),\:\left(x_2,\:y_2\right):\quad \sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$

RT = $=\sqrt{\left(1-1\right)^2+\left(-3-3\right)^2}$=$\sqrt{36}$

RT = 6

SU  $=\sqrt{\left(-2-4\right)^2+\left(-1-1\right)^2}$.

$=2\sqrt{10}$.

Diagonal RT is not congruent to Diagonal SU.

Therefore, Quadrilateral RSTU  is not a rectangle because the diagonals are not congruent.

So, it is False.