Chapter 1: Orienting Yourself: The Use of Coordinates

Exercise

Question

Question: 12 (i)

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(i) 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?

Answer

1
Find the distance of each point from the origin.
\text{OA = $\sqrt{(0 - 1)^2 + (0 - (-8))^2} = \sqrt{(-1)^2 + 8^2} = \sqrt{1 + 64} = \sqrt{65}$}
\text{OB = $\sqrt{(0 - (-4))^2 + (0 - 7)^2} = \sqrt{4^2 + (-7)^2} = \sqrt{16 + 49} = \sqrt{65}$}
\text{OC = $\sqrt{(0 - (-7))^2 + (0 - (-4))^2} = \sqrt{7^2 + 4^2} = \sqrt{49 + 16} = \sqrt{65}$}
2
Compare the three distances from the origin to the points. If they are equal, the points lie on the same circle centred at the origin.
\textbf{Answer: Yes. The points A, B, and C lie on circle K, whose radius is $\sqrt{65}$.}
\textbf{Reason: Each point is at the same distance, $\sqrt{65}$, from the origin O. Therefore, they lie on a circle centred at O with radius $\sqrt{65}$.}
Coordinate plane showing circle K centred at the origin, with points A, B, and C on its circumference.
Circle K centred at the origin with points A, B, and C on its circumference.
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