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Arcs

by Thouraya » Tue Mar 01, 2011 8:13 am
Hi,

When would an arc equal double the angle, and when would it equal the angle?

For ex, I have a circle with center O and angle POQ=90 degrees (with P and Q being on the circle). If we have O=90 degrees, and length of PQ is 4 pie, how can we find the radius?

Why do we consider PQ=90 not 180?

Thanks!
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by manpsingh87 » Tue Mar 01, 2011 8:21 am
Thouraya wrote:Hi,

When would an arc equal double the angle, and when would it equal the angle?

For ex, I have a circle with center O and angle POQ=90 degrees (with P and Q being on the circle). If we have O=90 degrees, and length of PQ is 4 pie, how can we find the radius?

Why do we consider PQ=90 not 180?

Thanks!
you can find the radius using formula l=2pir (x)/360

here x is an angle subtended by an arc at the center and in this case it is 90 hence;

4pi = 2pi r 90/360;
r=8
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by Thouraya » Tue Mar 01, 2011 8:35 am
Hi,

Thanks for your reply! Yes I noticed how they calculated the radius, but what i did not understand is why we assumed the angle 90? I thought that to find the arc, we multiple the angle by 2 hence 180?
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by Rich@VeritasPrep » Tue Mar 01, 2011 8:42 am
Hey Thouraya,

I'm wondering if you're actually thinking of the relationship between central angles and inscribed angles.

If you take two points on the circumference of a circle and create a central angle (i.e. with the center as the third vertex), the resulting angle will always be twice as large as an inscribed angle created using those same two points and a third point on the circumference.

Here's a diagram:
https://www.geom.uiuc.edu/~dwiggins/pict44.GIF

Is that what you were thinking of?
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by Thouraya » Tue Mar 01, 2011 8:49 am
Hi Raz,

I appreciate your help. Yes, I think I may be confused regarding the relation between arcs and central vs. inscribed angles.

Let's take the diagram as an example. Let's call one of the points (where green and red intersect) P and the other Q. And let us say that angle B=20 degrees, and angle A=30 degrees. Then in terms of arcs, how much would arc PBQ measure? vs PAQ?

This might help clarify actually.. Thanks!
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by Rich@VeritasPrep » Tue Mar 01, 2011 8:56 am
First, just to clarify, I'm assuming you're labeling angles A and B as 30 and 20 with the knowledge that they would NOT be an inscribed-angle, central-angle pair.

The diagram I linked to holds true for all cases.

But if you treat 30 and 20 as central angles, then all you have to do is figure out what fraction each number is out of 360.

30 is 1/12 of 360, therefore the arc resulting from a central angle of 30 will be 1/12 the full circumference of the circle.

Similarly, 20 is 1/18 of 360, therefore the arc resulting from a central angle of 30 will be 1/18 the full circumference of the circle.

Make sense?
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by Thouraya » Tue Mar 01, 2011 9:00 am
:) Yes, makes sense..

Let me put in clearer words what i dont understand.

We have two types of questions when it comes to arc: we are either asked about the measure or about the length.

To find the length, we just have to use the formula that u just mentioned: n/360 * circumference (where n is the measure of the arc)

HOWEVER,

when I am asked to find the MEASURE of the arc, I am confused how to calculate that. Usually, we are given a measure of an angle which is related to the arc, this angle is either inscribed or central. How does this inscribed vs. central angle affect how I calculate the measure of the arc? There's a formula where sometimes the measure of the arc is equal to the measure of this angle, other times, it is twice.

I hope Im making sense.. Thanks for your patience:)
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by Rich@VeritasPrep » Tue Mar 01, 2011 9:12 am
I think you might be confusing your terminology. An arc is by its very nature a length, and therefore the "measure of an arc" is the same thing as the length of an arc.

I also think that you're thinking of the basic definition of a radian. A radian is a central angle such that the resulting arc length is equal to the radius of the circle.

Luckily, you don't need to know trig for the GMAT :)
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