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