Secant line and diameter of a circle A secant line incident to a circle at points $A$ and $C$ intersects the circle's diameter at point $B$ with a $45^\circ$ angle. If the length of $AB$ is $1$ and the length of $BC$ is $7$, then what is the circle's radius?


The above diagram is almost self explanatory.  The perpendicular bisector of chord AC passes through the center O of the circle.  Since the diameter line makes an angle of 45 degrees with AC, angle MBO is 45 degrees and so triangle MBO is isosceles. Hence |MO|=3, |AM|=4 and by Pythagoras, r = |OA| =5.
