# FIND MISSING VERTEX OF 2D SHAPES IN COORDINATE GEOMETRY

## Properties of 2D Shapes of Diagonals

Properties of rectangle :

• Diagonals will have same length
• Midpoints of diagonals will be equal.

Properties of square :

• Diagonals are congruent
• Diagonals bisect each other.
• Diagonals are perpendicular.

Properties of rhombus :

• Diagonals bisect each other.
• Diagonals are perpendicular

Properties of parallelogram :

• Diagonals bisect each other.

Problem 1 :

Points P(4, 12), Q(9, 14), and R(13, 4) are three vertices of a rectangle. Find the coordinates of the fourth vertex S.

Solution:

Let the coordinate of the fourth vertex are S(a, b).

Midpoint of PR = Midpoint of QS

On comparing both sides,

 9+a2=1729+a=17a=8 14+b2=814+b=16b=2

So, the coordinates of the fourth vertex are (8, 2).

Problem 2 :

Find point D that makes ABCD a square where A(-1, 4), B(-2, -1), and C(3, -2).

Solution:

Let the coordinate of the fourth vertex are D(a, b).

Midpoint of AC = Midpoint of BD

 -2+a2=1-2+a=2a=4 -1+b2=1-1+b=2b=3

So, the coordinates of the fourth vertex are (4, 3).

Problem 3 :

Points A(-1, 2), B(-3, -3) and D(6, 3) form three corners of a parallelogram. Find the fourth vertex, C.

Solution:

Let the coordinate of the fourth vertex are C(a, b).

Midpoint of AC = Midpoint of BD

 -1+a2=32-1+a=3a=4 2+b2=02+b=0b=-2

So, the coordinates of the fourth vertex are (4, -2).

Problem 4 :

Find point P that makes PQRS a rectangle where Q(-1, -5), R(2, -3) and S(-2, 3).

Solution:

Let the coordinate of the fourth vertex are P(a, b).

Midpoint of PR = Midpoint of QS

 a+22=-32a+2=-3a=-5 b-32=-1b-3=-2b=1

So, the coordinates of the fourth vertex are (-5, 1).

Problem 5 :

Find point S that makes PQRS a rectangle where P(6, 36), Q(-30, 9), and R(-12, -15).

Solution:

Let the coordinate of the fourth vertex are S(a, b).

Midpoint of PR = Midpoint of QS

 -30+a2=-3-30+a=-6a=24 9+b2=2129+b=21b=12

So, the coordinates of the fourth vertex are (24, 12).

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