We draw in radii to
the x-axis and to the ends of the chord
Since the circle is tangent to the x-axis at (4,0), the y-coordinate
of the center is the same as the radius, r, so it's coordinates are
(4,r)
The triangle formed with the chord is isosceles since its two legs are
congruent radii. So we draw a median, in green, which is also an altitude,
a vertex angle bisector, and a perpendicular bisector which divides the
6-unit chord into two parts which are 3 units each. Also the green line
is 4 units long because it is the x-coordinate of the center.
Now we can use the Pythagorean theorem on either of the two right
triangles that the isosceles tringle was split into:
r² = 3² + 4²
r² = 9 + 16
r² = 25
r = 5
Since the center is r=5 and the center is (h,k) = (4,5), the equation is
(x - h)² + (y - k)² = r²
or
(x - 4)² + (y - 5)² = 5² or
(x - 4)² + (y - 5)² = 25
Edwin
