Solution #22

Situate the circle in the cartesian plane, using one chord as the x-axis and the other as the y-axis, and the point of intersection of the two chords as the origin. This gives four points on the circle: (-3,0), (4,0), (0,6), and (0,-2).

The center of the circle is at the same x-coordinate as the midpoint of the horizontal chord, and at the same y-coordinate as the midpoint of the vertical chord. Thus the center is at (1/2, 2).

We then calculate the distance from (1/2,2) to any point on the circle, such as (4,0). The distance is then √ ((4-1/2)2+ (2-0)2)  = √ (49/4+ 4) =√ (65/4)  = √ 65 /2. The diameter is then twice this distance, or √ 65 .