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.
