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charles.kao15
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Geometry 4th Edition Chapter 5, Problem Set, Q10

by charles.kao15 Fri Dec 18, 2015 8:40 pm

So not sure if this is errata or maybe I'm just not experienced in geometry (it's been over a decade since I've had to learn/use it).

The Geometry Book, 4th Edition, Chapter 5, Problem Set, Question 10.

“If angle ACB is 40 degrees (see figure), and the area of the circle is 81pi, how long is arc AXB?”

So looking at the diagram, when solving for what I thought was AXB (I thought 1/2 the arc of ACB), I got 1/2 of the results of the answer question. I did all the math correctly with the exception of interpretation of what AXB is. Below is essentially what happened:

Knowing area of the circle is 81pi, I solved for r in (pi)r^2. Revealing r = 9. So far so good.

I also know that the 40 degrees of ACB is only half of the angle measured at the radius, so angle of the arc from Point A to Point C at radius is 80 degrees. So far so good.

This is where interpretation goes haywire. When given AXB, and since X is a point between the A and C points, I figured AXB meant half the arc. So therefore I divided the angle by 2 (admittedly there is not enough data to state that X is half way between A and C).

So under the assumption that I had made previously, I took the circumference 2(pi)r = 18pi and multiplied by 40/360 to yield, ultimately, 2pi as the answer. But the answer key states 4pi because it states that AXB is the full arc (i.e the length from point A to point C as opposed to point A to point X).

Is this an errata or is it just a convention I've completely forgotten existed?
tommywallach
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Re: Geometry 4th Edition Chapter 5, Problem Set, Q10

by tommywallach Tue Dec 29, 2015 12:02 pm

Hey Charles,

I'll admit I don't totally understand your confusion. AXB is the arc from A, through, X, to B--if it were going to stop at X, it would just be called AX. So yes, it is the full 80 degrees. (The X is necessary, to differentiate from arc ACB, which would go the other way around).

-t
charles.kao15
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Re: Geometry 4th Edition Chapter 5, Problem Set, Q10

by charles.kao15 Wed Dec 30, 2015 9:04 pm

hi Tom,

Thanks for the comment, it makes sense to me. I'm not sure what I was thinking when I was first looking at the question. It makes sense now, why it wasn't making sense earlier....beats me.
tommywallach
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Re: Geometry 4th Edition Chapter 5, Problem Set, Q10

by tommywallach Mon Jan 04, 2016 6:06 pm

Ha! No worries. These things happen. The brain is a mysterious thing. :)