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

so here is the illustration of the problem

enter image description here

http://i1302.photobucket.com/albums/ag135/nktamc/th_Untitled_zpsc4a70930.png

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As this one cries to me "Do my homework" what did you try till now? –  Dominic Michaelis Mar 5 '13 at 9:27
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2 Answers

I'll model some of the problem-solving strategies and ways to approach problems like this, so in the future, if you post a problem, you can show some sketch of preliminary effort:

What we know:

Fact: The circumference of C the circle is $$C = \pi d = 2 \pi r $$

Given: where $r = $ the radius = $800$ feet $$\implies C = 2 \pi \times 800 = 1600\pi$$

Given: The distance measured between the two points of tangency on the circumference of the circle is given as $350$ feet.

Fact: The ratio of this distance to the circumference of the circle is equal to the ratio of the angle $\theta$ (the angle whose vertex is the origin of the circle and rays intersecting the points of tangency) to the angle measure of a circle = $2\pi$

Ratio (1) $$\text{ratio}\;(1) \;\;\dfrac{350}{C}\quad\text{ratio} (2)\;\; \dfrac {\theta}{2\pi}$$

$$\text{ratio}\;\; (1) = \text{ratio}\;\; (2) \implies\quad \dfrac{350}{C} = \dfrac{\theta}{2\pi}$$

Substituting $C$ from above with $r = 800$: $$\dfrac{350}{1600 \pi} = \frac{\theta}{2\pi} \iff \theta = \dfrac{350}{800}$$

This is the unknown angle from the center of the circle formed by radii joining the center of the circle to each of the points of tangency.

Now, knowing this, what can you conclude about the angle formed by the intersection of the two tangent lines?

  • It must be equal to $\pi - \theta = \pi - \dfrac{350}{800}$, in radians. Why?
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Bobatee: Is this clear now? Do you understand how finding the interior angle $\theta \implies$ the angle formed by the tangent lines is equal to $\pi - \theta$? –  amWhy Mar 5 '13 at 23:16
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I'd criticize your drawing but mine's probably worse ;)

To work this out, consider that the circumference of a circle is $2\pi r$ and the radians in a circle is $2\pi$. So then, with a radius of 800 ft, we know that two points a distance of $800\pi$ ft from each other will be parallel ($\pi$ radians between them). Given this, two points 350 ft away will be $\pi \frac{350}{800\pi} = \frac{350}{800}$ radians. However, this is the angle from the centre of the circle. The angle that you want (the angle between the tangent lines) will instead be $\pi - \frac{350}{800}$

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