Video Transcript
Determine the measure of arc
𝐶𝐵.
First, let’s see what we know. We know that the line segment 𝐶𝐴
and the line segment 𝐸𝐴 intersect outside the circle at point 𝐴. Both line segment 𝐶𝐴 and line
segment 𝐸𝐴 are secants of the circle. The figure has also told us that
the measure of arc 𝐶𝐵 is equal to the measure of arc 𝐸𝐷. We can also say that the sum of all
arc measures in the circle must be 360 degrees. We’re interested in finding the
measure of arc 𝐶𝐵. But in order to do that, we’ll need
to find the measure of arc 𝐵𝐷.
Based on what we said initially
about the intersection outside the circle, we can say that the angle made by two
intersecting lines outside the circle is half the positive difference of the
intercepted arcs, which is the intersecting secant angles theorem. The angle created by these
intersecting lines in our case is 34 degrees. 34 degrees is equal to the measure
of the intercepted arc 𝐶𝐸, which is 151 degrees, minus the measure of the
intercepted arc 𝐵𝐷 and then divided it by two.
We need to solve for the measure of
the arc 𝐵𝐷. So, we multiply both sides of the
equation by two, which gives us 68 equals 151 minus the measure of arc 𝐵𝐷. So, we subtract 151 from both
sides, and we get negative 83 degrees equals the negative measure of arc 𝐵𝐷. We multiply both sides of the
equation by negative one and flip the sides, and we get the measure of arc 𝐵𝐷
equals positive 83 degrees. But this is not our final
answer. We’re still trying to find the
measure of arc 𝐶𝐵.
But remember, we know that all of
these arcs must sum to 360 degrees. And we know that arc 𝐶𝐵 and arc
𝐸𝐷 must be equal. We could say that they’re equal to
𝑥 degrees. If we do that, we could then create
the equation 151 plus 𝑥 plus 𝑥 plus 83 equals 360. If we combine like terms, 151 plus
83 equals 234. 𝑥 plus 𝑥 equals two 𝑥. Therefore, 234 plus two 𝑥 equals
360 degrees. We subtract 234 from both sides of
the equation, which tells us two 𝑥 equals 126. Divide both sides by two, and we
see that 𝑥 equals 63. This tells us that both arc 𝐶𝐵
and arc 𝐸𝐷 is equal to a measure of 63 degrees. We were primarily interested in the
measure of arc 𝐶𝐵, and we found that that measure is 63 degrees.
