(A) Given the points A (1, -8), B (-4, 7) and C (-7, -4), show that they lie on a circle K whose centre is the origin O (0, 0). What is the radius of circle K?
(B) Given the points D (-5, 6) and E (0, 9), check whether D and E lie within the circle, on the circle, or outside the circle K.

A point lies on a circle with centre O and radius r iff its distance from O equals r.

$$\text{Use } OP = \sqrt{x^2 + y^2}.$$

(A) Compute OA, OB, OC.

$$OA = \sqrt{1^2 + (-8)^2} = \sqrt{1+64} = \sqrt{65}$$

$$OB = \sqrt{(-4)^2 + (7)^2} = \sqrt{16+49} = \sqrt{65}$$

$$OC = \sqrt{(-7)^2 + (-4)^2} = \sqrt{49+16} = \sqrt{65}$$

All three distances are equal, so A, B, C lie on a circle centred at O with radius $\sqrt{65}$ units (≈ 8.06).

(B) Check D and E against $\sqrt{65}$

$$OD = \sqrt{(-5)^2 + 6^2} = \sqrt{25+36} = \sqrt{61}$$

Since $\sqrt{61} < \sqrt{65}$, D lies inside K.

$$OE = \sqrt{0^2 + 9^2} = \sqrt{81} = 9.$$

Since $\sqrt{81} > \sqrt{65}$, E lies outside K.

### Concept Applied

A point lies on a circle with centre O and radius r if and only if its distance from O equals r; inside if less than r and outside if greater than r.