★ AIMS Think+ · Q12

Given:

The five points and centre O.

To Prove:

A, B, C lie on one circle; classify D and E.

Property / Theorem / Formula:

A circle is the set of points at a fixed distance from its centre.

$d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Proof:

${\mathrm{OA}}^2$ = 1 + 64 = 65. ${\mathrm{OB}}^2$ = 16 + 49 = 65. ${\mathrm{OC}}^2$ = 49 + 16 = 65.

All three have distance $\sqrt{65}$ from O, so they lie on that circle.

${\mathrm{OD}}^2$ = 25 + 36 = 61 < 65. ${\mathrm{OE}}^2$ = 81 > 65.

Results:

Radius $\sqrt{65}$ units; D is inside and E outside.