Video Transcript
Find 𝑥.
In this question, we’re asked to
find the value of 𝑥. And we can see that 𝑥 is the angle
between two tangent lines to our circle. That’s the line from 𝐴 to 𝐶 and
the line from 𝐴 to 𝐵. They just touch the circle at a
single point, so these are tangent lines. And we can find the value of 𝑥 by
recalling the following property for the angle between two tangent lines which
intersect at a point outside of our circle.
We recall the angle between two
tangent lines which intersect at a point is 180 degrees minus the measure of the arc
between the two points of tangency. In our diagram, the points of
tangency are the points 𝐵 and 𝐶. And the arc between 𝐵 and 𝐶 will
be the minor arc shown. And we know the measure of this
arc; its measure is 151 degrees. Then, our property tells us that
the value of 𝑥 is equal to 180 degrees minus the measure of arc 𝐵𝐶. So we can substitute the measure of
arc 𝐵𝐶, being 151 degrees, to get 𝑥 is equal to 180 degrees minus 151 degrees,
which we can calculate is 29 degrees.
Therefore, by using the fact that
the angle between two tangent lines which intersect at a point outside of a circle
is 180 degrees minus the measure of the arc between the two points of tangency, we
were able to show that 𝑥 is equal to 29 degrees.
Finally, let’s try and find the
angle between a tangent line and a secant line which intersect outside of a
circle. In this diagram, the tangent line
is 𝐴𝐷 and the secant line is 𝐶𝐵. And we want to find the measure of
the angle 𝐴𝐷𝐵. We’ll do this by using a very
similar method to the last three proofs. We’ll start by connecting 𝐴 and 𝐵
to construct a triangle 𝐴𝐵𝐷. We see that angle 𝐶𝐵𝐴 and angle
𝐴𝐵𝐷 are on a straight line, so their measures add to 180 degrees. So we have the measure of angle
𝐶𝐵𝐴 plus the measure of angle 𝐴𝐵𝐷 is equal to 180 degrees.
We then also have that the sum of
the measures of the internal angles in a triangle add to 180 degrees. So we have the measure of angle
𝐵𝐷𝐴 plus the measure of angle 𝐵𝐴𝐷 plus the measure of angle 𝐴𝐵𝐷 is equal to
180 degrees. And now we have two different
expressions which when added to the measure of angle 𝐴𝐵𝐷 is equal to 180
degrees. So these two expressions must be
equal. The measure of angle 𝐶𝐵𝐴 is
equal to the measure of angle 𝐵𝐷𝐴 added to the measure of angle 𝐵𝐴𝐷.
We can subtract the measure of
angle 𝐵𝐴𝐷 from both sides to find an expression for the measure of angle
𝐵𝐷𝐴. We have the measure of angle 𝐵𝐷𝐴
is equal to the measure of angle 𝐵𝐴𝐷 minus the measure of angle 𝐶𝐵𝐴. We can find an expression for the
measure of angle 𝐵𝐴𝐷 by first adding the following two radii to our diagram. And then we’ll use the fact that
the measure of the internal angles of quadrilateral 𝑀𝐴𝐷𝐵 add to 360 degrees. Since 𝐴 is a point of tangency for
our tangent line, angle 𝑀𝐴𝐷 is a right angle. So the sum of the internal angles
of this quadrilateral — the measure of angle 𝐴𝑀𝐵 plus 90 degrees plus the measure
of angle 𝐵𝐷𝐴 plus the measure of angle 𝐷𝐵𝑀 — is equal to 360 degrees.
We know the measure of the central
angle 𝐴𝑀𝐵 will be equal to the measure of the arc 𝐴𝐵. So we can substitute this into our
expression to give us the following. And by considering the internal
angles of triangle 𝐴𝐵𝐷, the internal angles sum to 180 degrees. So the measure of angle 𝐴𝐵𝐷 is
180 degrees minus the sum of the other two angles, the measure of angle 𝐵𝐴𝐷 and
the measure of angle 𝐵𝐷𝐴. Finally, since 𝑀𝐴 and 𝑀𝐵 are
radii, this means 𝑀𝐴𝐵 is an isosceles triangle. Therefore, the measure of angle
𝑀𝐴𝐵 and the measure of angle 𝑀𝐵𝐴 are equal. In particular, since angle 𝑀𝐴𝐷
is a right angle, we have the measure of angle 𝑀𝐴𝐵 is equal to 90 minus the
measure of angle 𝐵𝐴𝐷.
Now, all we need to use is the fact
that the measure of angle 𝐷𝐵𝑀 is the sum of the measure of angle 𝐴𝐵𝐷 and the
measure of angle 𝑀𝐴𝐵. We would substitute these into our
expression and then simplify. And we would be able to find the
following result. The measure of angle 𝐵𝐴𝐷 is
one-half the measure of the arc from 𝐴 to 𝐵. To do this, let’s clear some space
and go back to the following equation.
We can find an expression for the
measure of angle 𝐶𝐵𝐴 from our diagram. Angle 𝐶𝐵𝐴 is subtended by the
major arc from 𝐴 to 𝐶. And the measure of an inscribed
angle is one-half the measure of the arc it’s subtended by. So this is one-half the measure of
the arc from 𝐴 to 𝐶. We can substitute our expression
for the measure of angle 𝐵𝐴𝐷, giving us the following equation, which we can
rearrange for the measure of angle 𝐵𝐷𝐴, which gives us the following. The measure of angle 𝐵𝐷𝐴 is
one-half the measure of the major arc from 𝐴 to 𝐶 minus the measure of the arc
from 𝐴 to 𝐵.
An easy way to remember this is the
measure of the angle is one-half the difference of the measures of the two arcs
intercepted by the sides of the angle. And of course we take the positive
value for this difference.
Before we finish, there’s one more
property we can show. We’ve already shown the measure of
the angle between two tangents of a circle which meet at a point is 180 degrees
minus the measure of the minor arc between the two points of tangency. We can relate this to our other
results by considering the measure of the other arc; let’s call this 𝑦. These two arcs make up a full
circle, so the sum of their measures is 360 degrees. Subtracting 𝑥 from both sides of
the equation gives us 𝑦 is 360 degrees minus 𝑥. And we want to use this to consider
one-half the difference between these two arcs. That’s one-half 𝑦 minus 𝑥.
We’ll substitute this expression
for 𝑦 into one-half the difference. This gives us one-half 360 minus 𝑥
minus 𝑥, which if we simplify is 180 degrees minus 𝑥, which by using our first
result is the measure of angle 𝐴𝐶𝐵. In other words, we can also think
of the measure of the angle between two tangents which meet outside of a circle as
one-half the difference between the two arcs between the points of tangency.
