This is an exact method for finding the required radius of curvature. We'll actually find the equation of the circle passing through the 3 points.

In general, the x-value for the center of the circle passing through 3 points A (x1, y1), B (x2, y2), C (x3, y3), joined by lines with slopes m1 and m2 is given by:

`x_c=(m_1m_2(y_1-y_3)+m_2(x_1+x_2)-m_1(x_2+x_3))/(2(m_2-m_1)`

(The above formula is based on finding the intersection of the perpendicular bisectors of the 2 lines joining the 3 points, as follows.)

3 points circle

So for our given data points, we have:

`x_c=((2)(5)(1-8)+(5)(1+2)-(2)(2+3))/(2(5-2))=-10.833`

(We found the values for m1 and m2 in Method 2, above.)

Now to find the y-value for center of the circle. The formula for the first perpendicular bisector is given by:

`y_("Perp")=- 1/m_1(x-(x_1+x_2)/2)+(y_1+y_2)/2`

So for our data points, we can obtain the y-value for the center of the circle by substituting the known values, as follows:

`y_c=-1/2(-10.833-(1+2)/2)+(1+3)/2=8.1665`

So the center of the circle passing through the points `(1, 1), (2, 3)` and `(3, 8)` is given by `(-10.83, 8.17)`.

Finally, we can find the radius by simply finding the distance between the center of the circle and any one of the points on the circle. I have chosen `(1,1)`:

`text(Radius)=sqrt((x_2-x_1)^2+(y_2-y_1)^2`

`=sqrt((1-(-10.833))^2+(1-8.1665)^2)`

`=13.834`

[See more on Distance Formula.]

So the radius of curvature for the 3 points `(1, 1), (2, 3)` and `(3, 8)` is `13.83`, when finding the actual circle passing through the 3 points.

Our answer is slightly different to the answers obtained by using a parabolic model and linear approximations.

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