A circle has center 𝑀 and radius
𝑟 equals 21. Find the power of the point 𝐴 with
respect to the circle given that 𝐴𝑀 equals 25.
So, in this problem, what we’re
looking to find is the power of the point 𝐴. So in order to find out what this
is, we’re gonna be using the power of a point theorem. So, the first thing we’re 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, I’ve drawn a sketch.
So, in our sketch, we’ve got 𝐴,
which is our point. And then we’ve got 𝑀, which is the
center of our circle. I also put on 𝑟. And then, what I’ve also done is
I’ve drawn two points. So one is 𝐵 and one is 𝐶, 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 𝐴𝐵 multiplied by 𝐴𝐶. So, it’s 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 𝐶.
So, let’s have a look at what
information we’re given in the question. So, first of all, we know that the
radius is equal to 21. So, therefore, we can say that 𝐵𝑀
is equal to 21 and also 𝑀𝐶 is equal to 21. And we also know that 𝐴𝑀 is equal
to 25. Okay, great! So, we’ve found out what these
three values are. So now, let’s look at how we can
use our power of point theorem to find the power of the point 𝐴 with respect to the
circle. Well, first of all, we wanna find
out what 𝐴𝐵 is, and this is our section in pink.
Well, if you take a look at our
diagram, we can see that 𝐴𝐵 is going to be equal to 𝐴𝑀 minus our radius, or
minus 𝐵𝑀. So, therefore, 𝐴𝐵 is gonna be
equal to 25 minus 21. So, 𝐴𝐵 is equal to four. So we’ve now worked out 𝐴𝐵. So, now, what we need to work out
is 𝐴𝐶. Well, if we take a look at the line
𝐴𝐶, well, this is gonna be equal to 𝐴𝑀 plus 𝑀𝐶. So, therefore, it’s 𝐴𝑀 plus our
radius. So, therefore, 𝐴𝐶 is gonna be
equal to 25 plus 21. So, 𝐴𝐶 is gonna to be equal to
Okay, great! So, now, we’ve 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 𝐴 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 𝐴 with respect to the circle is
that-that is 184.