Question

If the points A(–2, –1), B(1, 0), C(a, 3) and D(1, b) form a parallelogram, find the values of a and b.

[Hint: Mid-point of AC = mid-point of BD.]

Answer 1

The diagonals of a parallelogram bisect each other, so mid-point of $$AC$$ = mid-point of $$BD$$.

Mid-point of $$AC$$: $$\left(\frac{-2 + a}{2},\frac{-1 + 3}{2}\right) = \left(\frac{a - 2}{2}, 1\right)$$

Mid-point of $$BD$$: $$\left(\frac{1 + 1}{2}, \frac{0 + b}{2}\right) = \left(1, \frac{b}{2}\right)$$

Equating the x-coordinates:

$$\frac{a - 2}{2} = 1$$

$$\Rightarrow a - 2 = 2$$

$$\Rightarrow a = 4$$

Equating the y-coordinates:

$$1 = \frac{b}{2}$$

$$\Rightarrow b = 2$$

Hence $$a = 4$$ and $$b = 2$$.