In a diagram, angle C is a right angle. If you draw the smallest possible circle through C and tangent to line AB,the circle will intersect line AC at J and line BC at K. What's the length of line JK.

Question

Line cb is 4 ab is 5 and AC is 3

Answer 1

The line PC is a diameter d of the circle, where P is the tangent point on AB. Let AP=x. In the right-angled triangle APC, x^2+d^2=AC^2=9; similarly, in CPB, (5-x)^2+d^2=16, because we have two right-angled triangles with common side PC. If we subtract the first equation from the second to eliminate d^2, we get 5(5-2x)=7, 25-10x=7 and 10x=18, so x=1.8. d^2=9-x^2=9-3.24=5.76, so d=2.4. Because the angle in a semicircle is a right angle, CJP=PKC and JPKC is a rectangle with JK=PC=d=2.4.