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 Post subject: GMAT Geometry
PostPosted: Mon Apr 29, 2013 8:11 am 
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In the figure above, ABCD is a rectangle inscribed in a circle. Angle AOD = 60° and the radius of the circle is 1. What is the ratio of the length of minor arc AD to the length of segment AD?
A. 3/π
B. 1/1
C. π/3
D. 9/8
E. π/2

(C) If you are really confused with this question, you may be able to get to the correct answer without doing any math. Certainly, minor arc AD is longer than line segment AD, but not by very much. So we can expect the answer to be slightly larger than 1. We can also expect the answer to contain π, since circumferences usually (but not always) have a π in them. Remember that π is approximately 3.14, so only choices (C), (D), and (E) are greater than 1. Choice (D) does not contain π, so it is probably not correct, and choice (E) is greater than 1.5, so it is probably too big.

This only leaves choice (C), which is correct here.
Now let’s actually solve the problem. Let’s start by determining the circumference of circle O:
Circumference = 2πr = 2π × 1 = 2π.

Because angle AOD measures 60°, we can deduce two things. First, the length of arc AD will be 60/360 = 1/6 of the total circumference of O. Second, if angle AOD measures 60° and the length of side AO is equal to the length of side OD, then all three angles of triangle AOD measure 60° and the triangle is an equilateral triangle with a side length of 1. Therefore, line segment AD has a length of 1.To determine the length of arc AD, multiply the circumference by 1/6:
ength of arc AD = 2π × 1/6= 2π/6 = π/3
The desired ratio is π/3 : 1, or π/3.

The correct answer is choice (C).
-------------

Perhaps an explanation for you to consider:
Calculate line segment AD.
Triangle ADC is right angled because, the diameter forms 90 at the circumference. From that you can figure out that on line AOC angle AOD and angle DOC are 60 and 120 respectively. Triangle COD is isoceles. ODC is 30 and therefore AOD is 60. Making AOD equilateral. Therefore segment AD is the radius =1. Very little calculation even though the explanation is long.

Arc AD is as you discussed above.
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 Post subject: Re: GMAT Geometry
PostPosted: Mon Apr 29, 2013 8:12 am 
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There are some weak statements in your reasoning. Let me go over it step-by-step:
"Triangle ADC is right angled because, the diameter forms 90 at the circumference."
It is true. Triangle ADC is a right triangle because it is inscribed in circle and one of its sides is diameter.

"From that you can figure out that on line AOC angle AOD and angle DOC are 60 and 120 respectively."
It is not clear why knowing the fact that triangle ADC is a right triangle gives us such information.
Angle DOC is 120⁰ because it is complementary with angle AOD. (angle DOC = 180⁰ – 60⁰ = 120⁰).
Conclusion that angle AOD is 60⁰ is not based on any facts at all.

"Triangle COD is isoceles. ODC is 30 and therefore AOD is 60."
Probably, you meant that angle ADO is 60⁰. If so then it is true. Triangle COD is isoceles because OC = OD as radii. angle ODC is 30⁰ because it equals (180⁰ – 120⁰)/2.

"Making AOD equilateral."
Remember, that as it turned out we still do not know anything about angle OAD so far. So at this point we could calculate it from right triangle ADC. Angle OAD = 90⁰ – 30⁰ = 60⁰. And then come to conclusion that AOD is equilateral.

But, the idea of your reasoning is possible though it takes much longer than original explanation.

In original explanation in order to show that triangle AOD is equilateral we use two facts:
- it is isosceles (two sides are equal as radii)
- one of its angles is 60 degrees
That is very simple and, probably, the fastest way to do.

If to follow you idea we will have much longer way to go in order to calculate all the angles in triangle AOD. We'll still be using isosceles triangle properties along the way.

So it is possible way to go but excessive. Also be careful not to make any mistakes along the way.

When taking GMAT simpler solutions will save you some time. But if it is more convenient for you to use longer way - why not, as long as you do not make a mistake and complete test on time.


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 Post subject: Re: GMAT Geometry
PostPosted: Mon Oct 21, 2013 12:39 pm 
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Joined: Tue Apr 13, 2010 8:48 am
Posts: 483
Hi,

I would like to confirm one thing.

If AOD = 60°, isn't there a rule that says the arc AD is twice this angle, so therefore Arc AD = 120°?

So (Alpha/360) × 2πr = (120/360) × 2π × 1 = (2π/3) and not (π/3).

Could you clarify why this rule does not apply here?

Thanks


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 Post subject: Re: GMAT Geometry
PostPosted: Mon Oct 21, 2013 12:41 pm 
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Joined: Fri Apr 09, 2010 2:11 pm
Posts: 459
Roquete wrote:
If AOD = 60, isn't there a rule that says the arc AD is twice this angle, so therefore Arc AD = 120?
This rule is about inscribed angles (formed by two chords). In our case we have a central angle and the degree measure of the arc is the same as of the angle.

Imagine breaking a circle into sectors of 60 degrees. There will be six such sectors, not three. Thus we divide the circumference, 2πr, by 6 to get πr/3 .
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