Determine the circumference of the circle
to the nearest hundredth.
The circumference of a circle is equal to
2 pi r which equals pi d,
where r is the radius of the circle
and d is the diameter.
So we want to use the given information
in order to determine the radius or diameter of the circle,
which we can then use to find the circumference.
So looking at the given information,
notice how this segment has one end point
at the center of the circle
and then it intersects this cord at a right angle.
And therefore this perpendicular segment bisects this cord,
meaning it cuts the cord into two congruent segments.
So the segment on the left is congruent
to the segment on the right.
So the total length of the cord is nine centimeters.
Going to the segment on the left would be 4.5 centimeters
and the same for the segment on the right,
which is also 4.5 centimeters.
Now from here because these two segments are perpendicular,
notice how we can form a right triangle
where this segment here would be the
hypotenuse of the right triangle.
But notice how this is also the radius of the circle.
So if we can find the length of this
hypotenuse of the right triangle,
it will give us the radius of the circle.
So let's label the hypotenuse r.
And also notice how we know the
lengths of two sides of this right triangle,
and therefore we can use the Pythagorean theorem
to find the length of the hypotenuse,
which again will give us the radius.
For the Pythagorean theorem is
a squared plus b squared equals c squared,
well a and b are the lengths of the two legs,
the legs are the two sides that form the right angle.
So this would be one leg,
this would be the other leg
a c is the length of the hypotenuse,
which is the side opposite the right angle
and also the longest side of the right triangle.
So let's let a equal 3, b equal 4.5 and c equal r.
And let's use the Pythagorean theorem in the form
c squared equals a squares plus b squared.
So again, c is the length of the hypotenuse which is r,
so we'd have r squared equals
a will be the leg that has the length of three centimeters,
so we have 3 squared plus
b will be the leg with the length of 4.5 centimeters,
so we'd have 4.5 squared.
So we have r squared equals
three squared is nine,
plus 4.5 squared is 20.25.
So we have r squared equals 29.25.
Now algebraically, you would have both a positive
and negative solution for r,
but because r is a length,
we're only concerned about the positive solution.
So we'll take the principal square root of both sides,
which gives us r equals the square root of 29.25.
Now that we know the length of the radius,
we can determine the circumference of the circle
and we'll leave the radius in it's exact form
and then we'll round the circumference
to the nearest hundredth.
So the circumference is equal to two pi r,
or in our case we have 2 times pi
times the square root of 29.25.
And now we'll use the calculator
to get our decimal approximation.
We have two
and then pi
and then times the square root
of 29.25.
Right arrow.
Enter.
To two decimal places this would be approximately
33.98 and the units would be centimeters.
I hope you found this helpful.