A circle's center is at #(3 ,1 )# and it passes through #(6 ,6 )#. What is the length of an arc covering #(7pi ) /12 # radians on the circle?
2 Answers
Explanation:
(WARNING: May be grossly worded, I will try to fix it later)
Okay so you have a circle. You want an arc length. Arc length is a portion of the circumference of a circle, so we need to find the circumference and then find the arc length (7
We know two points of the circle. We know the center (3,1), and we know a point the circle passes through (6,6). Since we know a point on the circle and the center, we can use them to find the radius, since a radius is any line on a circle that goes from the center to any point on the circle.
So we need to use the distance formula:
Plug in the coordinates, where (6,6) is our
Now we have the radius. Our goal is still to find the circumference. To get the circumference with the radius, we use the formula for circumference:
2
Plug in radius:
Okay, now we reached our first goal, which was to find the circumference. Now we can move on to finding the arc length.
Arc length is a fraction of the circumference. When they gave us the radians for the arc length, they gave us the fraction of the total radians for the circumference. A circle's total arc length in radians is 2
The fraction 7/12 from 7
We now have a fraction to use on our circumference. We want to know the arc length that takes up 7 parts of 24, so we multiply 7/24 by the circumference, the circumference being the total arc length.
If you simplify that, you will get the answer:
When the central angle
For this problem:
Explanation:
The circumference of a complete circle is
The length of an arc is a portion of the complete circumference, given by the ratio of the central angle
So, the arc length:
When we are working with radians, this simplifies to:
If we know the central angle, then we only need to find the radius of the circle, which is the distance from the center to any point on the circle.
In this problem, the central angle
We can use Pythagoras theorem to obtain this distance:
So: