Question

if A(-2,1), B(1,5), C(4,4), D(1,0) are the vertices of a quadrilateral, do the points form a parallelogram? Justify your answer.

Answer 1

We have that

A(-2,1), B(1,5), C(4,4), D(1,0)
using a graph tool
see the attached figure

we know that
A parallelogram is a simple quadrilateral with opposite sides parallel and equal in length
therefore
let's calculate the slope and distance of each side and compare them

side AB
slope m=(y2-y1)/(x2-x1)------> m=(5-1)/(1+2)----> m=4/3
distance AB=√4²+3³-------> dAB=5 units

side CD
slope m=(y2-y1)/(x2-x1)------> m=(0-4)/(1-4)----> m=4/3
distance CD=√4²+3³-------> dCD=5 units

side BC
slope m=(y2-y1)/(x2-x1)------> m=(4-5)/(4-1)----> m=-1/3
distance BC=√1²+3³-------> dBC=√10 units

side AD
slope m=(y2-y1)/(x2-x1)------> m=(0-1)/(1+2)----> m=-1/3
distance AD=√1²+3³-------> dAD=√10 units

(side BC and side AD are parallel)  and (side BC=side AD)
and
(side AB and side CD are parallel) and (side AB=side CD)

therefore

the answer is
Yes, it is a parallelogram