The coordinates of the vertices of quadrilateral abcd are a(4,1) b(1,5) c(-3,2) and d(0,-2). prove the quadrilateral is a square.

The quadrilateral is a SQUARE.

Step-by-step 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.

1,8,15,22,29,36,43,50

step-by-step explanation: