Simple problem in Euclidean Geometry -- Find the radius of a circle

Question

A student of mine brought the following question to my attention. I am currently not able to solve it, any help would be appreciated. It should be a simple circle theorem that I have now forgotten.

Question: Let $AB$ be a chord of a circle with centre $O$ and let $P$ be a point on $AB$. We are given that $OP$ has length $4$, $AP$ has length $11$, $BP$ has length $3$. We want to determine the radius of the circle.

Does this help?https://en.m.wikipedia.org/wiki/Intersecting_chords_theorem — Peter Szilas, May 13 '19 at 08:50

score 4 · Accepted Answer · answered May 13 '19 at 08:47

4

Let $M$ be the midpoint of $AB$. $AB=14$ and hence $PM=\dfrac{14}{2}-3=4$, which is equal to $OP$. So, $M$ is the same point as $O$. The radius is $7$.

Alternatively, we can use power of a point.

We have $(r-4)(r+4)=3\times 11$.

answered May 13 '19 at 08:47

CY Aries

23,393

CY Aries. Very nice. – Peter Szilas May 13 '19 at 09:06
@CY Aries -- Thank you for this! – AmorFati May 14 '19 at 00:30

Simple problem in Euclidean Geometry -- Find the radius of a circle

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