Question

a student attempts to draw a freehand circle. she marks 12 points on the circumference of the circle that she drew and finds their coordinates. this is shown in the diagram below.

(a) complete the table below showing the coordinates of the points starting with the point furthest to the right and moving in an anti-clockwise direction.

$x$	4.5				-2							3.5

(b) find the mean of the $x$-coordinates, $\bar{x}$, and the mean of the $y$-coordinates, $\bar{y}$.
(c) explain what the point $(\bar{x},\bar{y})$ represents in the context of the problem.

Explanation:

Part (a)

Step1: Identify points anti-clockwise

Start at (4.5, 0), list 12 points:

1. $(4.5, 0)$
2. $(3.5, 2)$
3. $(2, 3.5)$
4. $(0, 4.5)$
5. $(-2, 3.5)$
6. $(-3.5, 2)$
7. $(-4.5, 0)$
8. $(-3.5, -2)$
9. $(-2, -3.5)$
10. $(0, -4.5)$
11. $(2, -3.5)$
12. $(3.5, -2)$

Step2: Fill table with x-values

Match x to table columns:
$x: 4.5, 3.5, 2, 0, -2, -3.5, -4.5, -3.5, -2, 0, 2, 3.5$

Step3: Fill table with y-values

Match y to table columns:
$y: 0, 2, 3.5, 4.5, 3.5, 2, 0, -2, -3.5, -4.5, -3.5, -2$

Step1: Calculate $\bar{x}$

Sum x-values, divide by 12:
$$\bar{x} = \frac{4.5+3.5+2+0+(-2)+(-3.5)+(-4.5)+(-3.5)+(-2)+0+2+3.5}{12}$$
$$\bar{x} = \frac{0}{12} = 0$$

Step2: Calculate $\bar{y}$

Sum y-values, divide by 12:
$$\bar{y} = \frac{0+2+3.5+4.5+3.5+2+0+(-2)+(-3.5)+(-4.5)+(-3.5)+(-2)}{12}$$
$$\bar{y} = \frac{0}{12} = 0$$

The mean point $(\bar{x},\bar{y})$ is the average position of all marked points on the circle. For a perfect circle, this point corresponds to the center of the circle, as the coordinates of points on a circle are symmetric around its center.

Answer:

$x$	4.5	3.5	2	0	-2	-3.5	-4.5	-3.5	-2	0	2	3.5

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Part (b)