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.