Video Transcript
Find 𝑥.
Let’s look carefully at the diagram
we’ve been given. It consists of a circle. There are also two line segments
𝐴𝐸 and 𝐴𝐶, which are each segments of secants of this circle, because they each
intersect the circle in two places. The two secant segments intersect
one another at a point outside the circle, point 𝐴. And we’re given the measure of the
angle formed between the two secant segments. We’re asked to find the value of
𝑥, which we can see is the measure of the arc 𝐵𝐷. This is the minor intercepted arc
between the two secant segments.
The other information given on the
diagram is that the measure of the arc 𝐶𝐸 is 120 degrees. And this is the measure of the
major intercepted arc between the two secant segments.
To answer this problem, we need to
recall the intersecting secants theorem. This tells us that the angle
between two secants that intersect outside a circle is one-half the positive
difference of the measures of the arcs intercepted by the sides of the angle. We’ve already mentioned that the
arcs intercepted by the sides of the angle, that is, the line segments 𝐴𝐶 and
𝐴𝐸, are the arcs 𝐵𝐷 and 𝐶𝐸.
And so we can form an equation. We want the positive difference
between the measures of the arcs, so we need to subtract the measure of the minor
arc from the measure of the major arc. And we have 38 degrees is equal to
one-half the measure of the arc 𝐶𝐸 minus the measure of the arc 𝐵𝐷. We can then substitute 120 degrees
for the measure of the arc 𝐶𝐸 and 𝑥 degrees for the measure of the arc 𝐵𝐷. And we have 38 degrees is equal to
a half of 120 degrees minus 𝑥 degrees.
We can now solve this equation to
determine the value of 𝑥. First, we multiply each side of the
equation by two, giving 76 degrees is equal to 120 degrees minus 𝑥 degrees. We can then add 𝑥 degrees to each
side, so we have 𝑥 degrees plus 76 degrees is equal to 120 degrees, and finally
subtract 76 degrees from each side to give 𝑥 degrees is equal to 44 degrees. We’re just looking for the value of
𝑥, so this will be the numeric part of our answer.
By observing then that the line
segments 𝐴𝐶 and 𝐴𝐸 were secant segments and recalling the theorem concerning the
angle between two secants that intersect outside a circle, we found that the value
of 𝑥 is 44.
