Circle Problem

[DJBruce]

Here is a question for everyone:

 

Imagine a circle with line AB tangent to it and line AC being a cord of it. AC and AB form angle BAC with has a measure of 20 degrees. If arc AC is 2 units long what is the radius of the circle? This is not a homework problem simple something I came across.

 

Sorry my diagram is not the best.

 

[ATTACH]1878[/ATTACH]

[Sisyphus]

I'll tell you the answer if you give a convincing story of how you came across it somewhere other than homework. That way, even if you're lying, at least you worked for it.

[Kyrisch]

I won't do the problem for you, but I'll give you a hint. Add line segment CD where angle CDA is 90 degrees. That should make stuff a little clearer.

[DJBruce]

That is problem 15 from 1999 version of the Michigan Mathematical Prize Competition Part One.

 

As for the answer I feel like an idiot once you add the third leg. You use some trig to find the length of AB. Wow I spent forever not realizing that the length of AB was the radius.

[Willa]

That is problem 15 from 1999 version of the Michigan Mathematical Prize Competition Part One.

 

As for the answer I feel like an idiot once you add the third leg. You use some trig to find the length of AB. Wow I spent forever not realizing that the length of AB was the radius.

 

How can you find the length of AB...? It seems to me that B can be any arbitrary point along that line (AB).

[big314mp]

I think he means the line from the center of the circle to A or C??

 

In any case, it seems relatively straightforward:

 

http://mathworld.wolfram.com/CircularSegment.html

[Willa]

As for the answer I feel like an idiot once you add the third leg. You use some trig to find the length of AB.

 

Unless I'm grossly misinterpreting the problem, there's no need for trigonometry at all...

 

2 is the arc length, not the chord length, correct? After you find the angles of the triangle, you just need to use the formula for arc length (AKA application of basic circumference formula), which can be found in the link that big314mp posted. No sines or cosines anywhere.