Quadrilateral ARMY with A(p,q) R(0,0) M(r,0) and Y(p+r, q). Prove: ARMY is a parallelogram

Question

It can't be in a graph and has to be in the statement-reason format. I'll need the equations used as well

Answer 1

AR and MY are opposite sides.  To show that ARMY is a parallelogram, it is enough to show that AR and MY are both parallel and congruent.

 

Slope AR = (0-q)/(0-p) = q/p     Slope MY = (q-0)/(p+r-r) = q/p

 

Since AR and MY have the same slope, AR ll MY.

 

Length of AR = √[(p-0)² + (q-0)²] = √(p²+q²)

 

Length of MY = √[(p+r-r)² + (q-0)²] = √(p²+q²)

 

Since AR and MY have the same length, AR ≅ MY.

 

AR and MY are both parallel and congruent, so ARMY is a parallelogram.