Video Transcript
Find the measure of angle
𝐶𝐴𝐵.
Let’s begin by marking the angle
whose measure we’ve been asked to calculate on the diagram. We can then see that this is an
angle of tangency, as it is the angle between the tangent 𝐴𝐵 and the chord
𝐴𝐶. 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 𝐴𝐶 is the minor arc 𝐴𝐶. And if we sketch in the radii
connecting points 𝐴 and 𝐶 to the center of the circle, then the central angle
subtended by this arc is the angle marked in pink, angle 𝐴𝑀𝐶. So we have that the measure of
angle 𝐶𝐴𝐵 is one-half the measure of angle 𝐴𝑀𝐶.
Now we need to consider how to find
the measure of this angle. And to do so, we need to observe
that each side of triangle 𝐴𝐶𝐷 is the same length. So it is an equilateral
triangle. If we then sketch in the radius
connecting point 𝐷 to the center of the circle, we can conclude that the three
triangles into which we have divided the larger triangle 𝐴𝐶𝐷 are congruent. This is because each triangle has
two sides that are radii of the circle and one side from the original triangle.
The three triangles therefore have
three side lengths the same as each other and so are congruent by the side-side-side
congruency condition. This also means that the angles in
each triangle that are between the two sides that are radii of the circle are all
congruent. As angles around a point sum to 360
degrees, each of these angles is one-third of 360 degrees, which is 120 degrees. We therefore know that, in
particular, the measure of angle 𝐴𝑀𝐶 is 120 degrees. The measure of angle 𝐶𝐴𝐵 is
therefore one-half of 120 degrees, which is 60 degrees.
So, by identifying that angle
𝐶𝐴𝐵 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 𝐶𝐴𝐵 is 60 degrees.
