### Video Transcript

A circle has center π and radius
π equals 21. Find the power of the point π΄ with
respect to the circle given that π΄π equals 25.

So, in this problem, what weβre
looking to find is the power of the point π΄. So in order to find out what this
is, weβre gonna be using the power of a point theorem. So, the first thing weβre going to
do is have a look at the power of point theorem and see how we can apply it to solve
our problem. So in order to understand what the
power point theorem is, Iβve drawn a sketch.

So, in our sketch, weβve got π΄,
which is our point. And then weβve got π, which is the
center of our circle. I also put on π. And then, what Iβve also done is
Iβve drawn two points. So one is π΅ and one is πΆ, and
these are on the circumference of our circle. And what the power of a point
theorem states is that the power of a point is equal to π΄π΅ multiplied by π΄πΆ. So, itβs the distance from our
point to the edge of the circle multiplied by the distance from our point to the
other edge of the circle, so across the diameter to πΆ.

So, letβs have a look at what
information weβre given in the question. So, first of all, we know that the
radius is equal to 21. So, therefore, we can say that π΅π
is equal to 21 and also ππΆ is equal to 21. And we also know that π΄π is equal
to 25. Okay, great! So, weβve found out what these
three values are. So now, letβs look at how we can
use our power of point theorem to find the power of the point π΄ with respect to the
circle. Well, first of all, we wanna find
out what π΄π΅ is, and this is our section in pink.

Well, if you take a look at our
diagram, we can see that π΄π΅ is going to be equal to π΄π minus our radius, or
minus π΅π. So, therefore, π΄π΅ is gonna be
equal to 25 minus 21. So, π΄π΅ is equal to four. So weβve now worked out π΄π΅. So, now, what we need to work out
is π΄πΆ. Well, if we take a look at the line
π΄πΆ, well, this is gonna be equal to π΄π plus ππΆ. So, therefore, itβs π΄π plus our
radius. So, therefore, π΄πΆ is gonna be
equal to 25 plus 21. So, π΄πΆ is gonna to be equal to
46.

Okay, great! So, now, weβve got the two parts we
need. So, we can use them together to
find out what our power of point is. We can say, therefore, that the
power of the point π΄ is gonna be equal to four multiplied by 46. So, therefore, our power of point
is gonna be equal to 184. And we can work this out quickly
because four multiplied by 40 is 160. Four multiplied by six is 24. Add them together gives 184.

So, as I said, that gives us our
final answer, which is the power of the point π΄ with respect to the circle is
that-that is 184.