Question

What will be the value of a and b, if (-5, a), (-3, -3), (-b, 0) and (-3, 3) are the vertices of the parallelogram?

(a) a = 0, b = -1

(b) a = -1, b = 1

(c) a = 1, b = 1

(d) a = 0, b = 1

This question was posed to me in an international level competition.

The above asked question is from Geometry topic in section Coordinate Geometry of Mathematics – Class 10

Answer 1

Correct answer is (d) a = 0, b = 1

The best explanation: 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(-5, a) and R(-b, 0) and the ratio is 1:1

∴ x = \(\frac {1(-b)+1(-5)}{2} = \frac {-b-5}{2}\)

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

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(-3, -3) and S(-3, 3) and the ratio is 1:1

∴ x = \(\frac {1(-3)+1(-3)}{2} = \frac {-6}{2}\) = -3

y = \(\frac {1(3)+1(-3)}{2} = \frac {0}{2}\) = 0

Therefore, \(\frac {-b-5}{2}\) = -3

b = 1

\(\frac {a}{2}\) = 0

a = 0