# DETERMINING IF THE SET OF POINTS WILL BE A PARALLELOGRAM

To prove that the given points will create a parallelogram, we have to remember the following points.

• Length of opposite sides will be equal.
• Opposite sides will be parallel.
• Midpoints of the diagonals will be equal.

Problem 1 :

Verify that the quadrilateral with vertices P(-2, 2), Q(-2, -3), R(-5, -5) and S(-5, 0) is a parallelogram.

Solution :

Slope of PQ :

P(-2, 2) and Q(-2, -3)

x1 = -2, y1 = 2, x2 = -2, y2 = -3

Slope of QR :

Q(-2, -3) and  R(-5, -5)

x1 = -2, y1 = -3, x2 = -5, y2 = -5

Slope of RS :

R(-5, -5) and S(-5, 0)

x1 = -5, y1 = -5, x2 = -5, y2 = 0

Slope of SP :

S(-5, 0) and P(-2, 2)

x1 = -5, y1 = 0, x2 = -2, y2 = 2

The opposite sides PQ, RS and QR, SP have the same slope and are thus parallel to each other.

Midpoint of the diagonals :

Midpoint of diagonal PR :

P(-2, 2) and R(-5, -5)

Midpoint of S(-5, 0) and Q(-2, -3) :

Since the midpoints of the diagonal are equal and opposite sides are equal, it must be a parallelogram.

Problem 2 :

A quadrilateral has vertices K(-1, 4), L(2, 2), M(0, -1) and N(-3, 1). Verify that :

a) The quadrilateral is a square.

b) Each diagonal of the quadrilateral is the perpendicular bisector of the other diagonal.

c) The diagonals are equal in length.

Solution :

a)  Given, K(-1, 4), L(2, 2), M(0, -1) and N(-3, 1)

Length of  KL :

K(-1, 4) and L(2, 2)

x1 = -5, y1 = 0, x2 = -2, y2 = 2

KL = √[(-2 + 5)2 + (2 - 0)2]

= √[(3)2 + (2)2]

= √[9 + 4]

KL = √13

Length of  LM :

L(2, 2) and M(0, -1)

x1 = 2, y1 = 2, x2 = 0 , y2 = -1

LM = √[(0 - 2)2 + (-1 - 2)2]

= √[(-2)2 + (-3)2]

= √[4 + 9]

LM = √13

Length of  MN :

M(0, -1) and N(-3, 1)

x1 = 0, y1 = -1, x2 = -3 , y2 = 1

MN = √[(-3 - 0)2 + (1 + 1)2]

= √[(-3)2 + (2)2]

= √[9 + 4]

MN = √13

Length of  NK :

N(-3, 1) and K(-1, 4)

x1 = -3, y1 = 1, x2 = -1, y2 = 4

NK = √[(-1 + 3)2 + (4 - 1)2]

= √[(2)2 + (3)2]

= √[4 + 9]

NK = √13

KL = LM = MN = NK

c) Length of  KM :

K(-1, 4) and M(0, -1)

x1 = -1, y1 = 4, x2 = 0, y2 = -1

KM = √[(0 + 1)2 + (-1 - 4)2]

= √[(1)2 + (-5)2]

= √[1 + 25]

KM = √26

Length of  NL :

N(-3, 1) and L(2, 2)

x1 = -3, y1 = 1, x2 = 2, y2 = 2

NL = √[(2 + 3)2 + (2 - 1)2]

= √[(5)2 + (1)2]

= √[25 + 1]

NL = √26

KM = NL

So, diagonals are equal.

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