Question

What will be the coordinates of the fourth vertex S, if P(-1, -1), Q(2, 0), R(2, 3) are the three vertices of a parallelogram?

(a) (-5, -12)

(b) (5, -12)

(c) (5, 12)

(d) (-5, 12)

I have been asked this question in an international level competition.

I need to ask this question from Geometry topic in chapter Coordinate Geometry of Mathematics – Class 10

Answer 1

The correct choice is (c) (5, 12)

To explain: PQRS is a parallelogram. The opposite side of the parallelogram is equal and parallelogram. Also, the diagonals of the parallelogram bisect each other.

∴ O is the mid-point SQ and PR.

Midpoint of PR

Using, section formula x = \(\frac {mx_2+nx_1}{m+n}\) and y = \(\frac {my_2+ny_1}{m+n}\)

The points are P(-1, -1) and R(2, 3) and the ratio is 1:1

∴ x = \(\frac {1(-1)+1(2)}{2} = \frac {-1+2}{2} = \frac {1}{2}\)

y = \(\frac {1(3)+1(-1)}{2} = \frac {3-1}{2} = \frac {2}{2}\) = 1

Hence, the coordinates of O is (5, 6)

Midpoint of QS

Using, section formula x = \(\frac {mx_2+nx_1}{m+n}\) and y = \(\frac {my_2+ny_1}{m+n}\)

The points are Q(2, 0) and S(a, b) and the ratio is 1:1

∴ x = \(\frac {1(a)+1(2)}{2} = \frac {a+2}{2}\)

y = \(\frac {1(b)+1(0)}{2} = \frac {b}{2}\)

The coordinates of O is (5, 6)

Therefore, \(\frac {a+2}{2}\) = 5

a = 8

\(\frac {b}{2}\) = 6, b = 12

The coordinates of S are (5, 12).