### Video Transcript

The power of the points π΄, π΅, and πΆ with respect to the circle πΎ are π sub πΎ of π΄ equals four, π sub πΎ of π΅ equals 14, and π sub πΎ of πΆ equals negative one. The circle πΎ has center π and a radius of 10 centimeters. Calculate the distance between π and each of the points.

Letβs recall first how we calculate the power of a point with respect to a circle. For a circle πΎ centered at π with a radius of π units, the power of a point π΄ with respect to this circle is given by π sub πΎ of π΄ equals π΄π squared minus π squared, that is, the square of the distance between point π΄ and center of the circle minus the square of the radius. Weβve been given the power of three points π΄, π΅, and πΆ with respect to this circle πΎ, which has a radius of 10 centimeters. Letβs consider point π΄ first and substitute what we know into this formula.

The power of point π΄ with respect to this circle is four and the radius is 10. So we have the equation four equals π΄π squared minus 10 squared. Thatβs four equals π΄π squared minus 100. And then we can add 100 to each side of this equation, and we find that π΄π squared is equal to 104. π΄π is therefore the square root of 104. And we take only the positive value here as π΄π is a length. To simplify this surd or radical, we look for square factors of 104. And we find that 104 is equal to four times 26. π΄π is therefore the square root of four times 26. Thatβs the square root of four multiplied by the square root of 26, which is two root 26. So by rearranging the power of a point formula, we found the distance between point π΄ and the center of the circle.

Letβs now repeat this process for points π΅ and πΆ. The power of point π΅ with respect to the circle is 14, so we have the equation 14 is equal to π΅π squared minus 10 squared. That gives 14 equals π΅π squared minus 100. And adding 100 to each side, we find that π΅π squared is equal to 114. π΅π is then equal to the square root of 114. And as 114 has no square factors other than one, this value canβt be simplified.

Finally, letβs consider point πΆ. The power of point πΆ with respect to the circle is negative one. So we have the equation negative one equals πΆπ squared minus 10 squared. This can be rearranged to 99 is equal to πΆπ squared. And then taking the square root of each side of this equation, we find that πΆπ is equal to the square root of 99. Looking for square factors of 99, we recall that 99 is equal to nine multiplied by 11. So πΆπ is the square root of nine times 11, which is the square root of nine times the square root of 11, and this is equal to three root 11.

Weβve now found the distance between π and each of these three points. π΄π is equal to two root 26 centimeters. π΅π is equal to the square root of 114 centimeters. And πΆπ is equal to three root 11 centimeters. We can also deduce from the sign of the power of each point its position in relation to the circle. In general, if the power of a point π΄ with respect to a circle πΎ is positive, the point lies outside the circle. If the power of the point π΄ with respect to circle πΎ is equal to zero, then the point lies on the circumference of the circle, whereas if the power of point π΄ with respect to circle πΎ is negative, the point lies inside the circle.

As the power of points π΄ and π΅ is four and 14, which are both positive, these two points lie outside the circle, whereas as the power of point πΆ is negative one, which is obviously negative, this point lies inside the circle. We can further confirm this if we evaluate the length of π΄π, π΅π, and πΆπ as decimals. To one decimal place, they are 10.2, 10.7, and 9.9. As the lengths of π΄π and π΅π are each greater than 10, which is the radius of the circle, this confirms that points π΄ and π΅ are each outside the circle, whereas as the length of πΆπ is less than 10, this confirms that point πΆ is inside the circle. This is helpful for our understanding, but it wasnβt actually required in this question.

Our answer to the problem is that π΄π is equal to two root 26 centimeters. π΅π is equal to the square root of 114 centimeters. And πΆπ is equal to three root 11 centimeters.