Question Video: Finding the Measure of an Angle of Tangency Given the Measure of the Central Angle Subtended by the Same Arc Mathematics

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.