Question Video: Finding the Power of a Point Given the Radius and Distance from the Point to the Centre of the Circle Mathematics

Video Transcript

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 46.

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.