Answer:
The quadrilateral is a SQUARE.
Explanation:
Given
Four points
A(4,1)
B(1,5)
C(-3,2)
D(0, -2)
The quadrilateral formed will be ABCD with sides AB, BC, CD, AD. In order to prove if a quadrilateral is a square we have to prove that all sides of the quadrilateral are equal.
We will use the two-point distance formula to calculate lengths of sides of quadrilateral.
The distance formula:
d= √((x_2- x_1)^2+(y_2- y_1)^2 )
So, for side AB
AB= √((1- 4)^2+(5- 1)^2 )
= √((-3)^2+(4)^2 )
= √(9+16)
= √25
= 5 units
For BC
BC= √((-3- 1)^2+(2- 5)^2 )
= √((-4)^2+(-3)^2 )
= √(16+9)
= √25
= 5 units
For CD
BC= √((0-(-3))^2+(-2-2)^2 )
= √((0+3)^2+(-4)^2 )
= √(9+16)
= √25
= 5 units
For AD
AD= √((0-4)^2+(-2-1)^2 )
= √((-4)^2+(-3)^2 )
= √(16+9)
= √25
= 5 units
As all the sides are equal
AB=BC=CD=AD
= 5 units
So the quadrilateral is a square.
2 dot.
Explanation: