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:

missingverticesof2dshapeq1

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

Midpoint of PR = Midpoint of QS

4+132,12+42=9+a2,14+b2172,8=9+a2,14+b2

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       

missingverticesof2dshapeq2.png
-1+32,4-22=-2+a2,-1+b2(1,1)=-2+a2,-1+b2
-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+a2,2+b2=-3+62,-3+32-1+a2,2+b2=32,0
-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,b-32=-1-22,-5+32a+22,b-32=-32,-1
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

6-122,36-152=-30+a2,9+b2-3,212=-30+a2,9+b2
-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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