Question

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?

Answer 1

`QRST` is a rectangle

Explanation:

`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).`

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

Prove that:

1. 2 pairs of sides are parallel and one angle is 90°
2. 2 pairs of opposite sides are equal and one angle is 90°
3. 1 pair of sides is parallel and equal and one angle is 90°
4. All four angles are 90°
5. 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 `y` value `hArr` horizontal line
points S and T have the same `y` value `hArr` horizontal line
points Q and T have the same `x` value `hArr` vertical line
points R and S have the same `x` value `hArr` vertical line

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°

Answer 2

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

`OT=<4 1/2, -3 1/2>`

The vectors for the sides are

`QR`

`=OR -OQ=<4, 0>, and`, likewise,

`RS=<0, -5 1/2>, ST=<-4, 0> and TQ=<0, 5 1/2>`

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

Here, the opposite pairs of sides `QR=-ST and RS=-TQ`.

So, the figure is a parallelogram.

If one of the vertex angles is `pi/2`, QRST is a rectangle

The dot product `QR.RS=(4)(0)+(0)(-5 1/2)=0`.

So, QRST is a rectangle.

This method is applicable to any skew quadrilateral QRST.
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