HOW TO CHECK WHAT TYPE OF QUADRILATERAL FROM THE GIVEN POINTS

By remembering the following points helps us to prove the given points will create what type of quadrilateral.

Square :

All four sides will be equal and the diagonal will divide the square into two right triangles.

Rectangle :

In rectangles the opposite sides will be equal and the diagonal will divide the shape into two right triangles.

Rhombus :

Length of all sides will be equal and the diagonals will be perpendicular to each other.

Parallelogram :

Length of opposite sides will be equal and parallel. The midpoints point of the diagonals will be equal.

Trapezium :

Opposite sides will be parallel and other two sides will be non parallel.

Problem 1 :

The vertices of a quadrilateral are K(-1, 0), L(1, -2), M(4, 1) and N(2, 3). Verify that 

a) KLMN is a rectangle.

b) The lengths of the diagonals of KLMN are equal.

Solution :

verifying-geometric-properties-of-quadrilateral-with-analytic-geometry-s5

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

K(-1, 0) and  L(1, -2)

x1 = -1, y1 = 0, x2 = 1, y2 = -2

KL = √[(1 + 1)2 + (-2 - 0)2]

= √[(2)2 + (-2)2]

 = √[4 + 4]

KL =  √8

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

x1 = 1, y1 = -2, x2 = 4, y2 = 1

LM = √[(4 - 1)2 + (1 + 2)2]

= √[(3)2 + (3)2]

 = √[9 + 9]

LM =  √18

M(4, 1) and N(2, 3)

x1 = 4, y1 = 1, x2 = 2, y2 = 3

MN = √[(2 - 4)2 + (3 - 1)2]

= √[(-2)2 + (2)2]

 = √[4 + 4]

MN =  √8

N(2, 3) and  K(-1, 0)

x1 = 2, y1 = 3, x2 = -1, y2 = 0

NK = √[(-1 - 2)2 + (0 - 3)2]

= √[(-3)2 + (-3)2]

 = √[9 + 9]

NK =  √18

KL = MN and LM = NK. Opposite sides are equal.

Length of diagonal KM :

K(-1, 0) and M(4, 1)

KM = √[(4 + 1)2 + (1 - 0)2]

= √[(5)2 + (1)2]

 = √[25 + 1]

KM =  √26

Using Pythagorean theorem :

KM2 = KL2 + LM2

√262 = 82√182

26 = 8 + 18

26 = 26

Length of diagonal NL :  

N(2, 3) and L(1, -2)

NL = √[(1 - 2)2 + (-2 - 3)2]

= √[(-1)2 + (-5)2]

 = √[1 + 25]

NL =  √26

KM = NL

b) From the above calculation, the diagonals are equal.

Diagonals are also equal.

Problem 2 :

The vertices of a quadrilateral are A(0, 0), B(2, 3), C(5, 1) and D(3, -2).

a)  Check the type of quadrilateral

b) Verify that the diagonals of ABCD are perpendicular to each other. 

Solution :

verifying-geometric-properties-of-quadrilateral-with-analytic-geometry-s6

a)  Given, A(0, 0), B(2, 3), C(5, 1) and D(3, -2)

Length of AB :

A(0, 0) and B(2, 3)

AB = √[(2 - 0)2 + (3 - 0)2]

= √[(2)2 + (3)2]

 = √[4 + 9]

AB =  √13

Length of BC :

B(2, 3)and C(5, 1)

x1 = 2, y1 = 3, x2 = 5, y2 = 1

BC = √[(5 - 2)2 + (1 - 3)2]

= √[(3)2 + (-2)2]

 = √[9 + 4]

BC =  √13

Length of CD :

C(5, 1) and D(3, -2)

x1 = 5, y1 = 1, x2 = 3, y2 = -2

CD = √[(3 - 1)2 + (-2 - 1)2]

= √[(2)2 + (-3)2]

 = √[4 + 9]

CD =  √13

Length of DA :

D(3, -2) and A(0, 0)

x1 = 3, y1 = -2, x2 = 0, y2 = 0

DA = √[(0 - 3)2 + (0 + 2)2]

= √[(-3)2 + (2)2]

 = √(9 + 4)

DA =  √13

AB = BC = CD = DA all sides are equal. It may be a square or rhombus.

A(0, 0) and C(5, 1)

x1 = 0, y1 = 0, x2 = 5, y2 = 1

m = y2 - y1x2 - x1AC = 1 - 05 - 0AC = 15

B(2, 3) and D(3, -2)

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

m = y2 - y1x2 - x1BD = -2 - 33 - 2BD = -51BD = -5

Slope of AC x Slope of BD = (-1/5) x 5 

= -1

Since the diagonals are perpendicular to each other, it must a rhombus.

Problem 3 :

Verify that the quadrilateral with vertices O(0, 0), P(3, 5), Q(13, 7) and R(5, 1) is a trapezoid.

Solution :

In trapezoid, there will be two parallel sides and two non parallel sides.

verifying-geometric-properties-of-quadrilateral-with-analytic-geometry-s7

Given, O(0, 0), P(3, 5), Q(13, 7) and R(5, 1)

Slope of OP :

O(0, 0) and P(3, 5)

x1 = 0, y1 = 0, x2 = 3, y2 = 5

m = y2 - y1x2 - x1OP = 5 - 03 - 0OP = 53

Slope of PQ :

P(3, 5) and Q(13, 7)

x1 = 3, y1 = 5, x2 = 13, y2 = 7

m = y2 - y1x2 - x1PQ = 7 - 513 - 3PQ = 210 PQ = 15

Slope of QR :

Q(13, 7) and R(5, 1)

x1 = 13, y1 = 7, x2 = 5, y2 = 1

m = y2 - y1x2 - x1QR = 1 - 75 - 13QR = -6-8 QR = 34

Slope of RO :

R(5, 1) and O(0, 0)

x1 = 5, y1 = 1, x2 = 0, y2 = 0

m = y2 - y1x2 - x1RO = 0 - 10 - 5RO = -1-5 RO = 15

Here the slope of the line PQ and RO are equal.

PQ = RO = 1/5

Since the pair of sides are having same slope, the given coordinate will create parallelogram.

Problem 4 :

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

Solution :

verifying-geometric-properties-of-quadrilateral-with-analytic-geometry-s8

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

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

m = y2 - y1x2 - x1PQ = -3 - 2-2 + 2PQ = -5

Q(-2, -3) and  R(-5, -5)

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

m = y2 - y1x2 - x1QR = -5 + 3-5 + 2QR = -2-3QR = 23

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

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

m = y2 - y1x2 - x1RS = 0 + 5-5 + 5RS = 5

S(-5, 0) and P(-2, 2)

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

m = y2 - y1x2 - x1SP = 2 - 0-2 + 5SP = 23

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 10 :

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.Q

Solution :

verifying-geometric-properties-of-quatrilateral-with-analytic-geometry-s10

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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