Chords MN and PK intersect at point A so that MA = 3cm, NA = 16cm.

Chords MN and PK intersect at point A so that MA = 3cm, NA = 16cm. PA refers to KA as one to three (1/3). Find the PK and the radius of this circle.

By condition, the ratio of the segments PA and CA is known, from their ratio we find the CA:
PA / KA = 1/3;
KA = 3 * PA.
The intersecting chord theorem allows us to write the equality:
MA * NA = PA * KA = PA * 3 * PA
16 = PA²
PA = 4 (cm).
KA = 12 (cm).
Chord length PK:
4 + 12 = 16 (cm).
The condition does not say, probably you need to find the smallest radius of the circle.
The diameter of the circle is the largest chord, so it cannot be smaller than any of the other chords.
By condition, we are given two chords:
PK = 16 (cm);
MN = 19 (cm).
We get that the diameter is not more than 19 cm, respectively, the smallest radius will be 9.5 cm.
Answer: PK = 16 cm, the smallest radius is 9.5 cm.



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