Question

Given quadrilateral RSTU, determine if each pair of sides (if any) are parallel and which are perpendicular for the coordinates of the vertices.

R(-l, -5), S(8, 2), T(5, 5), U(-4, -2)

the options are
a) parallel
b) perpendicular
c) none

Answer 1

We would need to figure out the gradient of each "pair", as provided in the image.

The equation of the gradient is found by,

m = $\frac{ y_{2 } - y_{1 } }{ x_{2 } - x_{1 } }$

Parallel lines mean they have the same gradient
Perpendicular lines will follow the formula $m_{1 } * m_{2 } = -1$
"None" will apply to those that do not follow either.

Now we calculate the gradient of the pairs given to us by substituting the x and y values.

RS = $\frac{2-5}{8-(-1) }$
      = $-\frac{1}{3}$

TU = $\frac{-2 - 5}{-4 -5}$
      = $\frac{7}{9}$

Therefore, the "pair" RS and TU will be NONE.

ST = $\frac{5 - 2}{5-8}$
     = -1

RU = $\frac{-2 -5}{-4 -(-1) }$
      = $\frac{7}{3}$

Therefore, the "pair" ST and RU is also NONE.

Hope this helped! Feel free to ask me if there's any part of the working you don't understand :)

Answer 2

RS -- a) is parallel to -- TU (by same slope = 7/9)
ST -- a) is parallel to -- RU (by same slope = -1)

Therefore it is a parallelogram, not a rectangle, because there are two sets of parallel lines, but each.set is not perpendicular to the other set. 7/9 is not the negative reciprocal of -1. It would have needed to.be 7/9 and -9/7, or -1 and +1.