Find in radians the angle between the tangents to a circle at two points whose distance apart, measured on the circumference of the circle is 350 feet the radius of the circle being 800 ft.

If you can, please show an illustration. Thanks!

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A radian is defined as the angle of an arc the length of the radius of the circle. In your case the arc length is $^{350} / _{800} =\; ^7 / _{16}$ of the radius. From there the amount of radians should follow. – tesc Mar 4 '13 at 9:22
2(pi) = 360 degrees. => angle in radians = (arc length)/(radius of circle) = 350/800 = 7/16. – lsp Mar 4 '13 at 9:24

Perhaps you can draw a labelled picture. Let $O$ be the centre of the circle, let $P$ and $Q$ be the two points of tangency, and let $M$ be the point where the two tangents meet. It will turn out that the angle $POQ$ is not very far from $30$ degrees, so a realistic picture should have $\angle POQ$ of about that size.

One useful thing to remember is that the radius $OP$ is perpendicular to the tangent line $PM$ at $P$, and the radius $OQ$ is perpendicular to the tangent line at $Q$.

So the quadrilateral $OPMQ$ has two right angles. We have been asked for the angle at $M$. Since the angles of a (convex) quadrilateral add up to $360$ degrees, the angles at $M$ and at $O$ have sum $180$ degrees, that is, $\pi$ radians.

So the angle at $M$ is $\pi$ minus the angle at $O$.

The angle at $O$ is, in radians, $\dfrac{350}{800}$. This has been mentioned in the comments. There are various ways of remembering it. One way is to recall that all the way around the circle is a distance $(2\pi)(800)$. It is also a full rotation, so it is $2\pi$ radians. Thus $1$ foot along the circumference is $\dfrac{1}{800}$ of a radian, and therefore $350$ feet along the circumference is $\dfrac{350}{800}$ radians.

Finally, the angle at $M$ is therefore $\pi-\dfrac{350}{800}$ radians.

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