Question Video: Finding the Measure of an Arc in a Circle with Parallel Chords Mathematics • Third Year of Preparatory School

Video Transcript

Find the measure of the minor arc 𝐶𝐷, given that the measure of the minor arc 𝐴𝐷 is equal to 168 degrees and the measure of the major arc 𝐵𝐶 is 256 degrees.

The key to answering this question is to pay close attention to the distinction between a minor arc and a major arc between two points on any circle. For example, in the circle shown with two points 𝑋 and 𝑌, there are two arcs that could be called arc 𝑋𝑌. The smaller of the two is called the minor arc 𝑋𝑌, and the larger of the two is called the major arc 𝑋𝑌. We should also note here that when we’re dealing with the same two points, the sum of the measures of the major and minor arcs of these two points is 360 degrees. In the given problem, we see that the minor arc 𝐴𝐷 has a measure 168 degrees. And that’s where the minor arc is the smaller of the two arcs between the two points 𝐴 and 𝐷.

Similarly, we’re told that the measure of the major arc 𝐵𝐶 is 256 degrees. And this corresponds to the larger of the two arcs between the points 𝐵 and 𝐶 on the circle as shown. Now our unknown is the minor arc 𝐶𝐷. And before we do anything else, we should note that we have two parallel chords in our circle. And we know that the measure of arcs between parallel chords of a circle are equal. This means that the minor arcs 𝐶𝐷 and 𝐴𝐵 in the given circle are equal in measure.

So how can we calculate the measure of these minor arcs? Well, one way of doing this is to calculate the minor arc 𝐵𝐶. We know that the major arc 𝐵𝐶 has measure 256 degrees. Therefore, the minor arc 𝐵𝐶 will have measure 360 degrees minus 256 degrees. And that’s 104 degrees. And so the minor arc 𝐵𝐶 has measure 104 degrees.

Now remember, the minor arc 𝐴𝐷 has measure 168 degrees. So if we let the measure of the minor arcs 𝐴𝐵 and 𝐶𝐷 be lowercase 𝑎, then we see that the minor arc 𝐴𝐷 is made up of minor arc 𝐵𝐶, that’s 104 degrees, plus two multiplied by lowercase 𝑎. And we know that the minor arc 𝐴𝐷 has measured 168 degrees. So we can set up an equation as shown. We can solve this equation for 𝑎 by subtracting 104 degrees from both sides, which gives us two 𝑎 is 64 degrees. And if we then divide both sides by two, this gives us 𝑎 is 32 degrees. And this means that both minor arcs 𝐶𝐷 and 𝐴𝐵 have the measure 32 degrees. And this means that the measure of the minor arc 𝐶𝐷 is 32 degrees.

Now, if we wanted to go back and do an additional check, we could go back to the minor arc 𝐴𝐷 and find the major arc 𝐴𝐷. That will be 360 degrees minus 168 degrees, which is 192 degrees. And once we’ve labeled this on our diagram, we should find that 32 degrees, that’s 𝑎, minor arc 𝐴𝐵, plus 104 degrees, that’s minor arc 𝐵𝐶, plus 32 degrees, that’s 𝑎 again, which is minor arc 𝐶𝐷, plus 192 degrees, that’s major arc 𝐴𝐷, should equal 360 degrees, which it does. And this confirms that the measure of the minor arc 𝐶𝐷 is 32 degrees.