### Video Transcript

Two circles π and π intersect at
points π΄ and π΅, and the point πΆ satisfies πΆ belongs on the line π΅π΄ and πΆ does
not belong on the line segment π΅π΄. π· and πΈ are the points where the
line segment πΆπΈ intercects the circle π, and the line πΆπΉ is a tangent to
π. Given that πΆπ· equals seven and
π·πΈ equals 12, find π sub π of πΆ.

Thereβs an awful lot of information
given to us in the question, so letβs begin by drawing a diagram. We have two circles with centers π
and π. We donβt know which is larger. And actually, it doesnβt really
matter. These two circles intersect at the
points π΄ and π΅. Now weβre told that the point πΆ is
on the line π΅π΄, but it isnβt on the line segment π΅π΄. That means if we draw in the line
π΅π΄, πΆ is somewhere on this line, but it isnβt between π΄ and π΅. So perhaps itβs here. Thereβs then a line segment πΆπΈ,
which intersects the circle π at points π· and πΈ, so we can add this line segment
to our diagram. There is then a line πΆπΉ which is
a tangent to circle π. So, hereβs that final line. The last information weβre given is
the lengths of two line segments. πΆπ· is seven units and π·πΈ is 12
units.

So, we have a diagram. And now letβs look at what weβre
being asked to find. π sub π of πΆ means the power of
point πΆ with respect to circle π. Itβs calculated using the formula
πΆπ squared minus π squared, where π is the radius of the circle π. So itβs the distance between points
πΆ and the center of the circle squared minus the radius squared. However, we donβt have any of this
information, so weβre going to need a different approach. Letβs consider circle π first as
we have more information about this circle. In circle π, we know the lengths
of πΆπ· and π·πΈ, which are each segments of a secant to this circle. We can therefore recall the power
of a point theorem concerning the lengths of secant segments. This states the following.

Consider a circle π and a point πΆ
outside the circle. Let the line segment πΆπΈ be a
secant segment to the circle at π· and πΈ. Then the power of point πΆ with
respect to circle π is equal to πΆπ· multiplied by πΆπΈ. This is great because we know the
length of πΆπ·. Itβs seven units. And the length of πΆπΈ is seven
plus 12. Itβs 19 units. So we can work out the power of
point πΆ with respect to circle π as seven multiplied by 19, which is 133.

This isnβt what we were asked to
find though. We were asked to find the power of
point πΆ with respect to circle π. Well, the line at πΆπ΅ or π΅πΆ is
also a secant of the circle π, so it follows that the product of the lengths of the
secant segments πΆπ΄ and πΆπ΅ will also be equal to 133. This is because the power of a
point with respect to any given circle is always the same. So we have the equation πΆπ΄
multiplied by πΆπ΅ is 133. But this is useful because πΆπ΅
isnβt just a secant segment of circle π; itβs also a secant segment of circle
π. It is, in fact, a common secant
segment for the two circles.

So by the power of a point theorem
concerning the length of secant segments for circle π, we have that π π of πΆ is
equal to πΆπ΄ multiplied by πΆπ΅. And weβve just determined that this
will be equal to 133. In other words, what weβve found
then is that the power of point πΆ with respect to each circle found using their
common secant is the same. We can conclude then that the power
of point πΆ with respect to circle π is 133.