Question from Paul:
I have three known points. X1,Y1 = 0,0 X2,Y2 = 3,1 and X3,Y3 = 10,0. Given these three points, how do I find the height from the center of the 10' chord to the circle's circumference above?
If C is the centre of the circle the C has coordinates (5, k) for some k. (From the diagram I know that k must be negative.) Triangle SPC is a right triangle and hence from Pythagoras theorem
r2 = k2 + 52.
Since Q is on the circle the distance from Q to C is r and hence the distance formula gives
r2 = (5 - 3)2 + (k - 1)2
Solve the two equations for k and then find r.
Paul, start by drawing the arc.
You will see that the center of the circle must lie on the vertical
line x = 5, because that's half-way between points 1 and 3.
So we know the center point is (5, k) where k is the y value (clearly
The equation of a circle with radius r and center at (h, k) is
(x - h)² + (y - k)² = r²
Thus, we know that the origin (point 1) is on the circle. So we
substitute in (0, 0) for (x, y) and we get this:
(0 - 5)² + (0 - k)² = r²
r² = k² + 25
As well, we know that (3, 1) is on the circle - that's point 2. So:
(3 - 5)² + (1 - k)² = r²
r² = k² - 2k + 1 + 4
r² = k² - 2k + 5
Now we can combine these two equations, because both right hand sides
equal the same thing (r²):
k² + 25 = k² - 2k + 5
2k + 20 = 0
k = -10.
So the center is at (5, -10). Use the distance formula to find the
radius, then you will easily finish the problem.
Stephen La Rocque