Video Transcript
𝑀 is the center of a circle, and
𝐴, 𝐵, and 𝐶 are points on the circumference. If 𝐵𝐶 equals 13 centimeters and
the measure of angle 𝐶𝑀𝐵 is 84 degrees, find the area of the circle 𝑀, giving
the answer to the nearest square centimeter.
We know that the area of a circle
is 𝜋𝑟 squared. So really, this problem is about
finding the radius of this circle. Let’s begin by putting the
information we’ve been given on the diagram. 𝐵𝐶 is 13 centimeters, and the
measure of angle 𝐶𝑀𝐵 is 84 degrees. We don’t know the lengths of 𝑀𝐶
or 𝑀𝐵, but they’re each the radius of the circle.
Now, there are numerous different
approaches we could take. But one approach is to apply the
law of cosines in the triangle 𝐶𝑀𝐵. This states that 𝑎 squared equals
𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 cos 𝐴, where 𝑏 and 𝑐 represent two
sides of a triangle and 𝐴 represents the included angle. In our triangle, 𝑎 is 13
centimeters. The angle 𝐴 is 84 degrees. And the two sides which enclose
this angle 𝐴 are each the radius of the circle 𝑟.
We can therefore form an
equation. 13 squared equals 𝑟 squared plus
𝑟 squared minus two 𝑟 squared cos of 84 degrees. We can solve this equation to find
the value of 𝑟 squared, which we’ll then be able to substitute directly into our
area formula. Factorizing the right-hand side of
our equation by 𝑟 squared, we have 169 equals 𝑟 squared multiplied by two minus
two cos of 84 degrees. Dividing through, we have that 𝑟
squared is equal to 169 over two minus two cos 84 degrees. And we’ll keep our value for 𝑟
squared in this exact form.
We can then substitute this value
of 𝑟 squared into the area formula and evaluate on a calculator. Rounding our answer, and we have
that the area of circle 𝑀 to the nearest square centimeter is 296 square
centimeters.
As I mentioned, there are in fact
numerous approaches to this problem, which you can try out yourself if you wish. We could’ve applied the law of
sines in triangle 𝐶𝑀𝐵. Or we could’ve divided it in half
to form two right triangles and then used right-angle trigonometry.
