What is the circumference of the circle above?

Hey jatin2003
So when we know the angle is 60° and the length of the chord we can say that the radius would also be 6 coz a equilateral triangle is being formed with the n n 2 radii from the end of the chords to the center. So now that you know the radius, you can find out the circumference.
Let me know if this was clear for you.

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Hi Sumi747! Yes, the same concept, clicked in my mind, later. Thanks...!!

Does this help in any way ????

OQ = OS = Radius

Further drop a perpendicular from centre O to T on QS, where we have Triangle OQT = Triangle OTS
Further QT = TS = 3

And using Pythagorean triple we can find OT = 4 and OS = 5 = Radius

Now calculate the circumference.....

How the angle becomes 60 degrees, is it because arc is 1/6 of circumference and 360/6 = 60?? or something else, kindly explain

I am wondering if my approach is correct.

What is the circumference of the circle above?

(1) The length of arc QRS is one-sixth of the circumference.

arc QRS /circumference = 1/6

Insufficient b/c we don't know the arc length

(2) The length of chord QS is 6.

Notice that there is actually a smaller circle that can be formed and QRS represents one half of that circle.
A chord of 6 implies a radius of 3 for that smaller circle.

C = 2πr = 6π

6π/2 = 3π

Insufficient b/c we don't know how much of the circle 3π represents.

Combined:

3π is 1/6 of the circumference of the circle so the full circle is 18π

Sufficient.

We can calculate a part of the circumference with the following formula:
x°/360° * circumference = part of the circumference

1) not sufficient
2) not sufficient

1+2)
The part of the circumference (QRS) = 1/6*circumference
x°/360° * circumference = 1/6*circumference
-> x°/360° = 1/6
--> x = 1/6*360° = 60°

We know that QS = 6
-> Half of the chord = 3

If a triangle has a right angle we can use the ratios for triangles. We know that the angle of the triangle Q-center-S has 60° -> if we bisect this triangle we have 2 similar triangles. Each has a angle of 30° and the opposite side has a length of 3. We can use the ratio 3:4:5 for 30°-60°-90° triangles.
-> Radius = 5

Circumference = 2*pi*radius
-> we can calculate the circumference

--> 1+2) are sufficient

What is the circumference of the circle above?

(1) The length of arc QRS is one-sixth of the circumference.

we no nothing about the measurement of the circle. INSUFF.

(2) The length of chord QS is 6.

We no nothing about the arc. Image should not be assumed to represent the actual lengths. It could encompass 1/5th of the circle or 1/4 or 1/6th. The line could be bent at an angle or straight.

1 + 2:

Arc is 1/6 so the interior angle if we created a Sector on the circle is 60. The length of the chord is 6. This creates a triangle. Bisect the triangle from the vertices (from the center) to the chord (this creates a radius)
This is a 90-60-30 degree triangle. We can find the length of the radius. SUFF.

Hope this helps.

Statement 1 tells us that the corresponding ars, QRS, accounts for 1/6th of the circumference, implying that the angle enclosing the arc is 1/6th of the circle, i.e. 360/6 = 60 degrees.
Now, the nature of the triangle enclosed between the center of the circle, let's call it O, and points Q and S is such that angles at point Q, and point S are equal to each other, because sides, OQ and OS correspond to the radius of the circle.
From Statement 1, we know that the angle at the center of the circle is 60 degrees, therefore the other two angles can be calculated the following:
180-60=2x
x=60
Triangle 60-60-60 is an equilateral triangle ergo, sides are-6-6-6 (Statement 2), therefore the radius is of the circle is 6.
Circumference = 2rPi = pi*12