The circle 𝑀 in the figure has radius seven centimeters. Given that 𝐴𝐵 equals 14.9 centimeters, find the measure of angle 𝐴𝑀𝐶 and 𝑀𝐵, rounded to the nearest tenth.
So within this question, we’re asked to find two things, the measure of an angle and the length of a line segment. Let’s begin with the measure of the angle. 𝐴𝑀𝐶 is the angle formed travelling from 𝐴 to 𝑀 to 𝐶. So it’s this angle that I’ve marked here in orange. This angle is at the center of the circle with two other angles that have been marked, the right angle 𝐴𝑀𝐵 and the angle of 151 degrees, the angle 𝐶𝑀𝐵. The key fact that we need to remember in order to answer this part of the question is this: the sum of the measures of the central angles in a circle is 360 degrees.
So the sum of the three central angles, angle 𝐴𝑀𝐶, 90 degrees, and 151 degrees is 360 degrees. We have an equation that we can solve in order to find angle 𝐴𝑀𝐶. To find angle 𝐴𝑀𝐶, we need to subtract the other two angles from 360 degrees. Angle 𝐴𝑀𝐶 is equal to 360 degrees minus 90 degrees minus 151 degrees. It’s equal to 119 degrees. So we’ve answered the first part of the question.
Now let’s think about how to answer the second part. We’re given in the question two pieces of information about the lengths of various lines. 𝐴𝐵 is equal to 14.9 centimeters. And the radius of the circle is seven centimeters. The radius of a circle is a line segment whose endpoints are the center of the circle and a point on the circle itself. In this question, both 𝑀𝐴 and 𝑀𝐶 are radii of the circle. So now that we’ve marked all of the lengths onto the diagram, let’s think about how to calculate the length of the line 𝑀𝐵.
𝑀𝐵 is a side in the triangle 𝐴𝑀𝐵 in which we know the lengths of the other two sides. As this triangle is right angled, we can apply the Pythagorean theorem in order to calculate the length of the third side. The Pythagorean theorem tells us that in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. In this triangle, this means that 𝑀𝐵 squared plus seven squared is equal to 14.9 squared. And so we have an equation that we can solve in order to find the length of 𝑀𝐵.
Evaluating both seven squared and 14.9 squared gives 𝑀𝐵 squared plus 49 is equal to 222.01. Next, we can subtract 49 from both sides of the equation, which gives 𝑀𝐵 squared is equal to 173.01. To find 𝑀𝐵, we next need to square root both sides of the equation. And using a calculator to do this, gives 𝑀𝐵 is equal to 13.15332.
The question has asked for this length to the nearest tenth. And so we need to round our answer. Our answer to the problem then, the measure of angle 𝐴𝑀𝐶 is 119 degrees. And the length of 𝑀𝐵 is 13.2 centimeters, to the nearest tenth. Remember, the key fact we used at the start of this question was that the sum of the measures of the central angles in a circle is 360 degrees.