### Video Transcript

A circle with center π has a radius of 11 centimeters. Point π΄ lies eight centimeters from π and belongs to the chord π΅πΆ. Given that π΄π΅ equals three π΄πΆ, calculate the length of the line segment π΅πΆ, giving your answer to the nearest hundredth.

Letβs begin by drawing a diagram to help. We have a circle with its center at π and the radius of this circle is 11 centimeters. Thereβs also a chord π΅πΆ and point π΄ lies somewhere along this chord such that the length of π΄π΅ is three times the length of π΄πΆ. So, perhaps, point π΄ is here. We donβt know the length of π΄π΅ and π΄πΆ, but we do know the ratio between these two lengths. If π΄π΅ is three π΄πΆ, then if π΄πΆ is π₯ centimeters for some nonzero value of π₯, π΄π΅ will be three π₯ centimeters. We also know that point π΄ is eight centimeters away from point π, so we can add the length of this line segment to our diagram.

Weβre asked to calculate the length of the chord π΅πΆ, so we need to determine the value of this unknown π₯. The information weβre given consists of the length of line segments of the same chord, so we can recall a special case of the power of a point theorem. This states that let π΄ be a point inside circle π and let the line segment π΅πΆ be a chord passing through π΄. Then the negative of π sub π of π΄ is equal to π΄π΅ multiplied by π΄πΆ. The notation of π sub π of π΄ means the power of point π΄ with respect to circle π and is defined to be equal to π΄π squared minus π squared. Thatβs the square of the distance between points π΄ and π minus the square of the radius.

We know both of these lengths. π΄π is equal to eight and π is equal to 11. So, we can deduce that π sub π of π΄ is equal to eight squared minus 11 squared. Thatβs 64 minus 121 which is negative 57. We have then that negative negative 57, or simply 57, is equal to π΄π΅ multiplied by π΄πΆ. Remember, we wrote down expressions for the lengths of π΄π΅ and π΄πΆ involving an unknown π₯. π΄π΅ was defined to be three π₯ centimeters, and π΄πΆ was defined to be π₯ centimeters. So, we have the equation 57 is equal to three π₯ multiplied by π₯, which simplifies to 57 is equal to three π₯ squared.

We can now solve this equation to determine the value of π₯. First, we divide both sides of the equation by three, giving 19 is equal to π₯ squared. Next, we find the square root of each side of this equation, taking only the positive value as we require π΄π΅ and π΄πΆ to be positive because they are lengths. So, we find that π₯ is equal to the square root of 19.

Finally, we need to calculate the length of π΅πΆ, which is equal to the length of π΄π΅ plus the length of π΄πΆ, and thatβs three π₯ plus π₯, or simply four π₯. Weβve just determined that π₯ is equal to the square root of 19, so we can substitute this value for π₯. And we find that π΅πΆ is equal to four root 19. Weβre asked to give our answer to the nearest hundredth, so we need to evaluate this as a decimal. Itβs 17.4355 continuing, which to the nearest hundredth is 17.44.

So by recalling a special case of the power of a point theorem, we found that the length of the line segment π΅πΆ to the nearest hundredth is 17.44 centimeters.