Question Video: Finding the Angle of a Point outside a Circle given the Angle of the opposite Arc Mathematics • 11th Grade

Find 𝑥.

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Video Transcript

Find 𝑥.

Let’s have a look at the diagram we’ve been given more closely. It consists of a circle and then two lines 𝐴𝐶 and 𝐴𝐸. Each of these lines intersect with the circle in two places, which means the lines 𝐴𝐶 and 𝐴𝐸 are called secants. The value of 𝑥, which we’ve been asked to find, is the measure of the angle formed where these two secants meet.

We’ve also been given the measures of the two intercepted arcs formed by these two secants. The measure of the minor arc 𝐵𝐷 is 36 degrees. And the measure of the major arc 𝐶𝐸 is 132 degrees. We need to recall the relationship that exists between the angle formed by two secants and the measures of the two intercepted arcs. We recall then that “if two secants intersect outside a circle, the measure of the angle formed is half the difference of the measures of the intercepted arcs.”

Our two secants do intersect at a point outside the circle. They intersect at the point 𝐴. So we could apply this result. The measure of the angle formed is the value 𝑥 that we’re looking to find. We’re told that this is equal to half the difference of the measures of the intercepted arcs. So this is a half of the measure of the arc 𝐶𝐸 minus the measure of the arc 𝐵𝐷, both of which we’ve been given.

𝑥 is therefore equal to a half of 132 degrees minus 36 degrees. 132 minus 36 is 96. So 𝑥 is equal to a half of 96 degrees or a half times 96 degrees. A half of 96, which we can find by dividing 96 by two, is 48.

So we found that the value of 𝑥, which is the measure of the angle formed between the two secants 𝐴𝐶 and 𝐴𝐸, is 48 degrees.