# 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?

Nov 1, 2016

$Q R S T$ is a rectangle

#### Explanation:

$Q \left(4 \frac{1}{2} , 2\right) , R \left(8 \frac{1}{2} , 2\right) S \left(8 \frac{1}{2} , - 3 \frac{1}{2}\right) , \mathmr{and} T \left(4 \frac{1}{2} , - 3 \frac{1}{2}\right) .$

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 $\Leftrightarrow$ horizontal line
points S and T have the same $y$ value $\Leftrightarrow$ horizontal line
points Q and T have the same $x$ value $\Leftrightarrow$ vertical line
points R and S have the same $x$ value $\Leftrightarrow$ 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°

Nov 1, 2016

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

$O T = < 4 \frac{1}{2} , - 3 \frac{1}{2} >$

The vectors for the sides are

$Q R$

$= O R - O Q = < 4 , 0 > , \mathmr{and}$, likewise,

$R S = < 0 , - 5 \frac{1}{2} > , S T = < - 4 , 0 > \mathmr{and} T Q = < 0 , 5 \frac{1}{2} >$

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

Here, the opposite pairs of sides $Q R = - S T \mathmr{and} R S = - T Q$.

So, the figure is a parallelogram.

If one of the vertex angles is $\frac{\pi}{2}$, QRST is a rectangle

The dot product $Q R . R S = \left(4\right) \left(0\right) + \left(0\right) \left(- 5 \frac{1}{2}\right) = 0$.

So, QRST is a rectangle.

This method is applicable to any skew quadrilateral QRST.
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