What is the circumference of the circle? You may round your answer to one decimal place or leave it in terms of pi.

Question

Answer 1

We know the circumference of a circle is: c = 2*pi*r, so the only variable we are missing is r.

We can use either the Pythagorean theorem or the understanding of a 3-4-5 triangle to evaluate r.

Let us call the mid-point of string QM "H":

1. we know that QH = HM because H is the mid-point,
2. we know that QM = 24
3. So QM = QH + HM = QH + QH = 2*QH ----> 24 = 2*QH ----> QH = 12

Now, we have 2 of the 3 sides of our right triangle.

1. to use the Pythagorean theorem we know that c = squareroot (a^2 + b^2 ), where c = r.
2. Plugging in we see that r = squareroot (12^2 + 9^2) = squareroot ( 144 + 81) = squareroot (225) = 15
3. to use 3-4-5
4. We know that 3*3 = 9
5. And that 3*4 = 12
6. And therefore the third side of our right triangle must be 3 * 5 = 15

We now take the knowledge that r = 15 and plug this into the circumference formula, such that:

1. c = 2*pi*15 = 30*pi

Answer 2

The line from P to the chord QM is a perpendicular bisector of the chord. Assume the point of intersection is T. Therefore the base of the right angle triangle QPT is QT = 24/2 = 12, The radius of the circle is segment PQ which is also the hypotenuse of the right angle triangle PQT such that PQ² = PT² + QT², PQ² = 9²+12², and

PQ² = 81 + 144 = 225, PQ = radius= √225 = 15 , and the circumference of the circle C = 2πr = 2(15)π = 30π