P is a point inside a circle and A is a point on the circumference.Find the radius of the circle.

Question

P is a point inside a circle and A is a point on the circumference. The minimum distance between A and P is 2 cm and the maximum distance between A and P is 8 cm. Find the radius of the circle. I think the radius must be greater than 4 cm. But next how to proceed ??

Answer 1

You can use triangular inequality to show that the minimum and maximum distances are attained when the point $A$ is on the diameter passing through $P$ (see Figure in the bottom).

For example, $$A'P+OP = r,$$ where $r$ is the circle radius, and, by triangular inequality on triangle $AOP$, $$AP+OP \geq r,$$ so that $$AP \geq A'P.$$ So no matter where you take $A$ the distance from $P$ will be greater than that of $A'$ from $P$. Thus $$A'P=2\ \mbox{cm}.$$ Similarly, you can express $A''P$ as $$A''P = r + OP,$$ whereas, again for triangular inequality on $AOP$, $$AP \leq r + OP,$$ which yields $$ AP \leq A''P.$$ In conclusion $A''P = 8$ cm and the radius is $r = \frac{A'P + A''P}{2}=5$ cm.

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Answer 2

The minimum distance from the point on the circumference will be at a right angle to the edge of the circle and the maximum distance will pass through the center point and also be at a right angle. Adding to two distances will make the diameter of the circle: $8+2 = 10$cm, therefore the radius is $5$ cm.