Question

Verify that the following points taken in order to form the vertices of a rhombus

A(3, −2), B(7, 6), C(−1, 2) and D(−5, −6)

Answer 1

Distance = ${\displaystyle \sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}}$

AB = ${\displaystyle \sqrt{{(7-3)}^2+{(6+2)}^2}}$

= ${\displaystyle \sqrt{{\left(4\right)}^2+{\left(8\right)}^2}}$

= ${\displaystyle \sqrt{16+64}}$

= ${\displaystyle \sqrt{80}}$

BC = ${\displaystyle \sqrt{{(-1-7)}^2+{(2-6)}^2}}$

= ${\displaystyle \sqrt{{(-8)}^2+{(-4)}^2}}$

= ${\displaystyle \sqrt{64+16}}$

= ${\displaystyle \sqrt{80}}$

CD = ${\displaystyle \sqrt{{(-5+1)}^2+{(-6-2)}^2}}$

= ${\displaystyle \sqrt{{(-4)}^2+{(-8)}^2}}$

= ${\displaystyle \sqrt{16+64}}$

= ${\displaystyle \sqrt{80}}$

AD = ${\displaystyle \sqrt{{(-5-3)}^2+{(-6+2)}^2}}$

= ${\displaystyle \sqrt{{(-8)}^2+{(-4)}^2}}$

= ${\displaystyle \sqrt{64+16}}$

= ${\displaystyle \sqrt{80}}$

AB = BC = CD = AD = ${\displaystyle \sqrt{80}}$

All the four sides are equal.

∴ ABCD is a rhombus.