# Polygon QRST has vertices #Q(4 1/2, 2), R (8 1/2, 2) S(8 1/2, -3 1/2), and T (4 1/2, -3 1/2).# ls polygon QRST a rectangle?

##### 2 Answers

#### Explanation:

To decide whether this is a rectangle or not, we have the following options to choose from:

Prove that:

- 2 pairs of sides are parallel and one angle is 90°
- 2 pairs of opposite sides are equal and one angle is 90°
- 1 pair of sides is parallel and equal and one angle is 90°
- All four angles are 90°
- The diagonals are equal and bisect each other. (same midpoint)

I will go with option 1, because this only requires finding the slope of each of the 4 lines.

Note that:

points Q and R have the same

points S and T have the same

points Q and T have the same

points R and S have the same

Therefore QRST has to be a rectangle because horizontal and vertical lines meet at 90°.

The opposite sides are therefore parallel and equal and the angles are 90°

See explanation.

#### Explanation:

The position vectors to the vertices are

#OQ=<4 1/2, 2>,OR=<8 1/2, 2>, OS=<8 1/2>, -31/2> and

The vectors for the sides are

Use vectors V and kV are ( like or unlike ) parallel vectors.

Here, the opposite pairs of sides

So, the figure is a parallelogram.

If one of the vertex angles is

The dot product

So, QRST is a rectangle.

This method is applicable to any skew quadrilateral QRST.

,