Video Transcript
Find the measure of angle
𝐵𝐴𝐶.
Let’s begin by marking the angle
whose measure we’ve been asked to find on the diagram.
We can now see that we’ve been
asked to calculate an angle of tangency, because the marked angle is the angle
between the tangent 𝐴𝐶 and the chord 𝐴𝐵. We can therefore recall the
following theorem. The measure of an angle of tangency
is equal to half the measure of the central angle subtended by the same arc. The arc that connects the endpoints
of the chord 𝐴𝐵 is the minor arc 𝐴𝐵, and the central angle subtended by this arc
is angle 𝐴𝑀𝐵.
So we have that the measure of
angle 𝐵𝐴𝐶 is half the measure of the angle 𝐴𝑀𝐵. The angle 𝐴𝑀𝐵 is marked on the
diagram with a small square, indicating that it is a right angle. So its measure is 90 degrees. The measure of angle 𝐵𝐴𝐶 is
therefore equal to one-half of 90 degrees, which is 45 degrees.
So, by identifying that angle
𝐵𝐴𝐶 is an angle of tangency and then recalling that the measure of an angle of
tangency is half the measure of the central angle subtended by the same arc, we’ve
found that the measure of angle 𝐵𝐴𝐶 is 45 degrees.
