Video Transcript
Determine the measure of the arc 𝐶𝐵.
Let’s have a closer look at the diagram we’ve been given. We have a circle and the line segments 𝐴𝐶 and 𝐴𝐸. Now, these are each segments of secants of the circle because they each intersect the circle in two places and pass outside the circle. These two secant segments intersect at a point outside the circle, point 𝐴. And we’ve been given the measure of the angle formed between them.
The other information we’re given is the measure of the arc 𝐶𝐸, which is the larger of the two arcs intercepted by these two secant segments. Now, we’re asked to determine the measure of the arc 𝐶𝐵. The only thing we know about this arc is that it is the same length as the arc 𝐸𝐷. Let’s think about how we can use the fact that we know the angle formed between these two secant segments to help us.
Well, the intersecting secant theorem tells us that the angle between two secants or secant segments that intersect outside a circle is half the positive difference of the measures of the arcs intercepted by the sides of the angle. The arcs intercepted by the sides of the angle at 𝐴 are 𝐵𝐷 and 𝐶𝐸. 𝐶𝐸 is clearly the larger of these two arcs, so we can form an equation. 34 degrees is equal to one-half the measure of the arc 𝐶𝐸 minus the measure of the arc 𝐵𝐷. Now, we know the measure of the arc 𝐶𝐸; it’s 151 degrees. So we could substitute this value into the equation and then solve it to find the measure of the arc 𝐵𝐷. But how will this help us?
Well, we know that the measure of the entire circumference of a circle is 360 degrees. So the measure of the arc 𝐶𝐸 plus the measure of the arc 𝐵𝐷 plus the measure of the arc 𝐶𝐵 plus the measure of the arc 𝐷𝐸 must be 360 degrees. We know the measure of the arc 𝐶𝐸. We’ve just discussed how we can find the measure of the arc 𝐵𝐷. And it’s the measure of the arc 𝐶𝐵 we want to find. The measure of the arc 𝐷𝐸, remember, is the same as the measure of the arc 𝐶𝐵. So in fact, we have one less unknown than we thought.
We can change our equation to the measure of the arc 𝐶𝐸 plus the measure of the arc 𝐵𝐷 plus twice the measure of the arc 𝐶𝐵 is equal to 360 degrees. And now we see that once we’ve determined the measure of the arc 𝐵𝐷, we’ll be able to find the measure of the arc 𝐶𝐵. Returning to our earlier equation then, we can multiply both sides by two, which will eliminate the fraction on the right-hand side and give 68 on the left-hand side. We can also substitute 151 degrees for the measure of the arc 𝐶𝐸. And we have 68 degrees equals 151 degrees minus the measure of the arc 𝐵𝐷. We can add the measure of the arc 𝐵𝐷 to each side of this equation and then subtract 68 degrees from each side. And we find that the measure of the arc 𝐵𝐷 is 83 degrees.
We can now substitute the measures of the arcs 𝐶𝐸 and 𝐵𝐷 into our second equation. And we have 151 degrees plus 83 degrees plus twice the measure of the arc 𝐶𝐵 is equal to 360 degrees. Subtracting 151 and 83 degrees from each side of this equation, we find that twice the measure of the arc 𝐶𝐵 is 126 degrees. Finally, we can divide both sides of this equation by two to give the measure of the arc 𝐶𝐵 is 63 degrees. So, by recalling the angles between intersecting secants theorem and the fact that the measure of the entire circumference of a circle is 360 degrees, we found that the measure of the arc 𝐶𝐵 is 63 degrees.
