Calculate Radii

Radii are half the diameters: $R = 50/2 = 25$ cm and $r = 34/2 = 17$ cm.

$R = 25, r = 17$

Find distance from center to chord

The line segment $AP$ is a chord of the smaller circle with length 16 cm. The perpendicular distance $d$ from the center to this chord bisects it ($8$ cm) and forms a right triangle with radius $r$.

$d = \sqrt{17^2 - 8^2} = \sqrt{289 - 64} = 15$

Calculate length of CF

The distance $d$ is also the distance from the center to chord $CF$ of the larger circle. Using the Pythagorean theorem for the larger circle: $CF/2 = \sqrt{R^2 - d^2}$.

$CF = 2 \times \sqrt{25^2 - 15^2} = 2 \times 20 = 40$