In the figure above, ABCD is a rectangle inscribed in a circle. Angle

OA = OD = Radius
i.e. Angle OAD = Angle ODA = (180-60)/2 = 60
i.e. Triangle OAD is an Equilateral Triangle with side = Radius = r

i.e. Length of Line AD = r

and Length of Minor Arc AD = (60/360)2πr = πr/3

Ratio of Minor Arc AD / Length of Line AD = (πr/3) / r = π/3

Answer: Option C

Circumference = \(2\pi*r = 2\pi\)
Minor Arc AD = \(\frac{60^{\circ}}{360^{\circ}}*2\pi = \frac{\pi}{3}\)

If radius = \(1\) and \(\angle\)AOC = \(60\), then all sides of \(\triangle\)AOC are equal to 1.
Thus, \(\frac{\pi}{3} : 1\) and answer choice C

Triangle OCD is isosceles. OD=OC = 1 (Radius); Hence angles opposite to equal sides equal, therefore Angle ODC = Angle OCD.

Angle AOD is exterior angle of triangle OCD.

Angle ODC + Angle OCD = Angle AOD; since OCD = ODC, we have angle OCD=ODC = 30Degrees.

Further, triangle ADC is right with Angle ACD = 30. and forms a 30-60-90 right triangle. We have AC=2 Dia circle.

Therefore AD = 1 (Rules of 30-60-90)

Minor Arc AD = (60/360) x 2 x pi x 1 = Pi/3.
AD = 1

So ratio is Pi/3.

Hope this is a correct approach.

The answer is pi/3. the triangle is an equilateral triangle with each side r and arc is one sixth of the circumference

800score Official Solution:

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).

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