The chords MN and PK intersect at point A so that MA = 3 cm, NA = 16 cm, PA: KA = 1: 3.

The chords MN and PK intersect at point A so that MA = 3 cm, NA = 16 cm, PA: KA = 1: 3. Find the length of the chord PK.

By the condition of the problem, the ratio of the segments of the chord of the RK is equal to 1/3.
RA / KA = 1/3 → KA = 3 * RA.
According to the intersecting chord theorem, we write the equality:
MA * NA = PA * KA
3 * 16 = PA * 3 * PA
16 = PA²
PA = 4 (cm).
KA = 3 * 4 = 12 (cm).
The length of the chord RK is equal to:
4 + 12 = 16 (cm).
Answer: the length of the RK chord is 16 cm.