Video Transcript
Find 𝑥.
In this question, we’re asked to
find the value of 𝑥. And we can see in our diagram that
𝑥 is the angle between two secant lines which intersect outside of our circle. And we can find the measure of 𝑥
by recalling the following fact. The angle between two secant lines
in a circle which intersect outside of a circle is one-half the positive difference
of the measures of the arcs intercepted by the sides of the angle.
To apply this property, let’s do
this step by step. First, let’s mark the sides of the
angle of 𝑥. We can see that 𝑥 is the angle
between the lines 𝐴𝐶 and 𝐴𝐸. So the two sides of our angle are
the line segment 𝐴𝐶 and the line segment 𝐴𝐸. Next, we need to find the measures
of the arcs intercepted by the two sides of our angle. The first side of our angle
intersects the circle at the point 𝐵, and the second side of our angle intersects
the circle at the point 𝐷. So one of the arcs we’re going to
use is the arc from 𝐵 to 𝐷. Similarly, the first side of our
angle intercepts the circle at the point 𝐶, and the second side of our angle
intercepts the circle at the point 𝐸. So the other arc we’re interested
in is the arc from 𝐶 to 𝐸.
Finally, the measure of our angle
will be one-half the positive difference between the measures of these two arcs. And since the arc from 𝐶 to 𝐸 is
bigger than the arc from 𝐵 to 𝐷, this gives us the following result. 𝑥 will be equal to one-half
multiplied by the measure of arc 𝐶𝐸 minus the measure of arc 𝐵𝐷. And we’re given both of these
values in the diagram. The measure of arc 𝐶𝐸 is 132
degrees, and the measure of arc 𝐵𝐷 is 36 degrees. So we substitute these values into
our formula. We get that 𝑥 is equal to one-half
multiplied by 132 degrees minus 36 degrees. And we can then evaluate this
expression. 132 minus 36 is equal to 96. And if we multiply this by
one-half, we get 48. Therefore, 𝑥 is equal to 48
degrees.
