Video Transcript
Find ๐ฅ.
Letโs look carefully at the diagram
weโve been given. It consists of a circle. There are also two line segments
๐ด๐ธ and ๐ด๐ถ, 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 ๐ด. And weโre given the measure of the
angle formed between the two secant segments. Weโre asked to find the value of
๐ฅ, which we can see is the measure of the arc ๐ต๐ท. 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 ๐ถ๐ธ 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. Weโve already mentioned that the
arcs intercepted by the sides of the angle, that is, the line segments ๐ด๐ถ and
๐ด๐ธ, are the arcs ๐ต๐ท and ๐ถ๐ธ.
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 ๐ถ๐ธ minus the measure of the arc ๐ต๐ท. We can then substitute 120 degrees
for the measure of the arc ๐ถ๐ธ and ๐ฅ degrees for the measure of the arc ๐ต๐ท. And we have 38 degrees is equal to
a half of 120 degrees minus ๐ฅ degrees.
We can now solve this equation to
determine the value of ๐ฅ. First, we multiply each side of the
equation by two, giving 76 degrees is equal to 120 degrees minus ๐ฅ degrees. We can then add ๐ฅ degrees to each
side, so we have ๐ฅ degrees plus 76 degrees is equal to 120 degrees, and finally
subtract 76 degrees from each side to give ๐ฅ degrees is equal to 44 degrees. Weโre just looking for the value of
๐ฅ, so this will be the numeric part of our answer.
By observing then that the line
segments ๐ด๐ถ and ๐ด๐ธ were secant segments and recalling the theorem concerning the
angle between two secants that intersect outside a circle, we found that the value
of ๐ฅ is 44.
