Question Video: Finding the Measure of an Angle using the Relationship Between Tangents, Radii and Angles in a Semicircle Mathematics

Video Transcript

Given that 𝐵𝐶 is a tangent to the circle with center 𝑀 and the measure of angle 𝐴𝑀𝐷 is 97 degrees, find the measure of angle 𝐶𝐵𝐷.

We know that in any circle, a tangent and radius or tangent and diameter meet at 90 degrees. We are told in the question that the measure of angle 𝐴𝑀𝐷 is 97 degrees. We need to calculate the measure of the angle 𝐶𝐵𝐷. The alternate segment theorem states that, in any circle, the angle between a chord and a tangent through one of the endpoints of the chord is equal to the angle in the alternate segment. In this question, the measure of angle 𝐶𝐵𝐷 is equal to angle 𝐵𝐴𝐷.

Let’s now consider how we can calculate the angle 𝐵𝐴𝐷 using the information on the diagram. Triangle 𝑀𝐴𝐷 is isosceles as the line segment 𝑀𝐴 is equal to the line segment 𝑀𝐷. They are both radii of the circle. This means that the angles inside the triangle, angle 𝑀𝐴𝐷 and angle 𝑀𝐷𝐴, are equal. These can be calculated by subtracting 97 from 180 and then dividing by two. 180 minus 97 is equal to 83. Dividing this by two or halving it gives us 41.5. Angles 𝑀𝐴𝐷 and 𝑀𝐷𝐴 are both equal to 41.5 degrees. As the angle 𝑀𝐴𝐷 is the same as the angle 𝐵𝐴𝐷, we can see, using the alternate segment theorem, that the measure of angle 𝐶𝐵𝐷 is also 41.5 degrees.