### Video Transcript

A point is at a distance 40 from the centre of a circle. If its power with respect to the circle is 81, what is the radius of the circle rounded to the nearest integer?

The power of a point is a measure of the distance from that point to a circle. Suppose 𝐴 represents the coordinates of the point, 𝐶 represents the coordinates of the centre of the circle, and 𝑟 represents the radius of the circle. We then have the following equation: the power of the point 𝐴 with respect to the circle 𝐶 is defined as the distance between the point 𝐴 and the centre 𝐶 squared minus the radius of the circle squared.

In this question, we’ve been given the distance from the centre of the circle to the point 𝐴𝐶, which is 40. And we’ve been given the power of the point, which is 81. We’re asked to calculate the radius of the circle. Substituting the power of the point and the distance between the point and the centre of the circle, we have the equation 81 is equal to 40 squared minus 𝑟 squared.

We now want to solve this equation to find the radius of the circle. As 𝑟 squared currently has a negative coefficient, negative one, I’ll begin by adding it to both sides of the equation, giving 𝑟 squared plus 81 is equal to 40 squared. Next, I’ll subtract 81 from both sides, giving 𝑟 squared is equal to 40 squared minus 81. 40 squared is equal to 1600 and subtracting 81 gives 1519.

To find the value of 𝑟, we need to take the square root of both sides of the equation. I’m only taking the positive square root here as the radius of a circle is positive by definition. Evaluating this square root using a calculator gives the decimal 38.97435.

The question asked for our answer to be given to the nearest integer. So rounding this value, we have that the radius of the circle is 39.