Circle 𝑀 has the radius of 12
centimeters where the length of 𝐶𝐵 is 16 centimeters. Find the length of arc 𝐶𝐵 giving
the answer to two decimal places.
First, let’s put all the
information we’re given onto the diagram. We’re told the circle has a radius
of 12 centimeters. So the lines 𝑀𝐶 and 𝑀𝐵 are each
12 centimeters. We’re also told that the length of
the line segment 𝐶𝐵 is 16 centimeters. What we’re looking to calculate in
this question is the length of the arc 𝐶𝐵, the part that I’ve marked in pink.
So in order to do this, we need to
know the size of the central angle, the angle that’s been marked as 𝜃 in the
diagram. Now we haven’t been given angle 𝜃,
so we’re going to need to calculate it from the other information in the
question. What you notice is that the 𝜃 lies
inside the triangle 𝑀𝐵𝐶, and we know the lengths of all three sides. They’re 12 centimeters, 12
centimeters and 16 centimeters.
If we know the lengths of all three
sides of a triangle, then we can calculate the size of any of its angles using the
law of cosines. So the law of cosines for an angle,
using the letters in this question, tells us that cos of 𝜃 is equal to 𝐵𝑀 squared
plus 𝐶𝑀 squared minus 𝐵𝐶 squared over two multiplied by 𝐵𝑀 multiplied by
𝐶𝑀. So let’s substitute the values for
each of these lengths. This tells us that cos of 𝜃 is
equal to 12 squared plus 12 squared minus 16 squared over two multiplied by 12
multiplied by 12. Evaluating each of these tells us
that cos of 𝜃 is equal to 32 over 288.
Now to find the value of 𝜃, we
need to use the inverse cosine function. 𝜃 is equal to cosine inverse of 32
over 288. Evaluating this on my calculator
tells me that 𝜃 is equal to 83.62062 dot dot dot. Now I’m going to keep this value on
my calculator screen because I need to use it in the next stage of my calculation,
and I don’t want to make my answer inaccurate by introducing any rounding
So the next stage of this question
is we need to actually calculate the length of this arc 𝐶𝐵. So the arc length can be found by
finding the circumference of the full circle two 𝜋𝑟 and then multiply it by the
portion of the circle that we have. So that’s 𝜃 over 360. So that’s why keeping this value on
my calculator screen has been really useful cause now I can use it exactly in this
stage of the calculation. I have 83.62062 divided by 360, and
then multiplied by two times 𝜋 times the radius of circle which is 12.
So if I evaluate all of this on my
calculator, I have a value of 17.513463. And if I return to the question, it
asks me to give my answer to two decimal places. So rounding my answer and including
the units for an arc length, which in this case are centimeters, tells me that the
length of arc 𝐶𝐵 is 17.51 centimeters.