Verifying a Shape

Question

I've tried this question repeatedly? Can any person who knows the analytic geometry field give me an answer? HELP?
Quadrilateral PQRS has vertices P(3, 2), Q(0, 4), R(-2, 1) and S(1, -1)
a) Verify that quadrilateral PQRS is a square.
b) Find the perimeter of the rectangle

Answer 1

Juan,

 

This question has been answered in previous posting(s).  Try working the suggested solution(s) and then ask questions if you get stuck at a particular step.

Answer 2

I suggest the following:

 

(1) Plot the 4 points.  You will note that the 4 sides are PQ, QR, RS, and SP.

(2) Use the distance formula to calculate the lengths of the 4 sides.

 

L = √[(x₁ - x₂)² + (y₁ - y₂)²],

 

where (x₁,y₁) and (x₂,y₂) are the end points of a side.  To be a square, all 4 lengths must be equal.  Are they equal?

 

(3) To be a rectangle or square, adjacent sides must also be perpendicular.  At each end point, where adjacent sides intersect, you have 2 lines parallel to the 2 adjacent sides, intersecting at the end point.  Find the slopes m and m' of the 2 lines that intersect at the end point of 2 adjacent slides.

 

m = (y₂ - y₁) / (x₂ - x₁)

 

If

 

mm' = -1,

 

the adjacent sides are perpendicular.

 

If you end up with 4 sides of equal length and all adjacent sides are perpendicular, then you have a square.

Answer 3

I gave you this answer a few days ago.  Please stop posting this question.