Question Video: Using Arc Measures to Determine the Relationship Between to Chords in a Circle Mathematics

Video Transcript

In the diagram, the measure of arc 𝐴𝐵 is equal to 62 degrees, the measure of arc 𝐵𝐶 equals 110 degrees, and the measure of arc 𝐴𝐷 equals 126 degrees. What can we conclude about the line segments 𝐴𝐷 and 𝐵𝐶? (A) They are parallel. (B) They are neither parallel nor perpendicular. (C) They are perpendicular. (D) They’re the same length. (E) They are parallel and of the same length.

Let’s begin by adding the measure of each of our arcs to the diagram. We’re told the measure of arc 𝐴𝐵 equals 62 degrees, the measure of arc 𝐵𝐶 equals 110, and the measure of arc 𝐴𝐷 equals 126. Since the measure of each arc is the angle that arc makes at the center of the circle, it follows that the sum of all arc measures that make up that circle is 360 degrees. And this is really useful because it will allow us to calculate the measure of arc 𝐶𝐷.

We say that the sum of the arc measures is 360 degrees. And then we can replace the various arc measures with their values. So the measure of arc 𝐴𝐵 is 62, the measure of arc 𝐵𝐶 is 110, and so on. This left-hand side simplifies to 298 degrees plus the measure of arc 𝐶𝐷. And then we can find that measure of arc 𝐶𝐷 by subtracting 298 from both sides. It’s 360 minus 298, which is of course 62 degrees.

So why is this useful? Well, we know that the measure of arcs between parallel chords of a circle are equal, and the opposite is also true. That is, if the measure of two arcs between two distinct chords is equal, those chords must in fact be parallel. We in fact have that the measure of arc 𝐴𝐵 is equal to the measure of arc 𝐶𝐷. And so that must mean that line segments 𝐴𝐷 and 𝐵𝐶 are in fact parallel. We’re therefore able to disregard options (B), (C), and (D). And so we need to choose between option (A) and option (E), where option (A) is that they are parallel and option (E) is that they’re not only parallel, but they’re of the same length.

Well, if the chords are parallel and equal in length, then the arcs between the endpoints of the chords will be equal in measure. But we can see that the measure of arc 𝐵𝐶 is not equal to the measure of arc 𝐴𝐷. They are in fact 110 and 126 degrees, respectively. So 𝐵𝐶 and 𝐴𝐷 cannot be of equal length. And so the answer is (A). They’re parallel.