Video Transcript
The point 𝐴 is outside a circle
with center 𝑀. The line between 𝐴 and 𝐶
intersects the circle at 𝐵 and 𝐶, and the line between 𝐴 and 𝐸 meets the circle
at points 𝐷 and 𝐸. Given that the measure of angle
𝐶𝑀𝐸 equals 130 degrees and the measure of angle 𝐵𝑀𝐷 is equal to 56 degrees,
find the measure of angle 𝐴.
We’ve been given a description of
our circle and lines, but we haven’t been given an image. The best place to start here is
with sketching. We know we have a circle and that
point 𝐴 is outside the circle. We also know that there is a line
between 𝐴 and 𝐶; there’s a line with endpoints 𝐴 and 𝐶. And this line intersects the circle
at 𝐵 and 𝐶. If we draw a line from 𝐴 to the
circle, we know that the endpoints of this line were 𝐴 and 𝐶, and the other
intersection along the circle was point 𝐵.
Similarly, we have a line between
𝐴 and 𝐸 that meets the circle at 𝐷 and 𝐸. We’ll draw another line from point
𝐴. The endpoint is 𝐸 and its other
intersection point is 𝐷. The circle has a center 𝑀. We’ve been told the measure of
angle 𝐶𝑀𝐸, which would be this angle, is 130 degrees. And we’ve been told the measure of
angle 𝐵𝑀𝐷, which would be this angle, and that measures 56 degrees. And we want to know the measure of
angle 𝐴.
Now, of course, when we look at our
sketch, we know that these angles are a little bit off. But this sketch gives us enough
information to figure out how we’re going to try and solve for the measure of angle
𝐴. Because we have two lines
intersecting outside of a circle, then the angle created by the two lines
intersecting outside the circle is half the positive difference between the
intercepted arcs.
And that means the measure of angle
𝐴 is the angle created outside the circle. It’s going to be one-half the
measure of arc 𝐶𝐸 minus the measure of arc 𝐷𝐵. Arc 𝐶𝐸 measures 130 degrees; arc
𝐵𝐷 measures 56 degrees. 130 minus 56 equal 74, and half of
74 is 37. And so, a circle under these
conditions will have the angle measure 𝐴 equal to 37 degrees.
