A circle with center 𝑀 has a radius of 11 centimeters. Point 𝐴 lies eight centimeters from 𝑀 and belongs to the chord 𝐵𝐶. Given that 𝐴𝐵 equals three 𝐴𝐶, calculate the length of the line segment 𝐵𝐶, giving your answer to the nearest hundredth.
Let’s begin by drawing a diagram to help. We have a circle with its center at 𝑀 and the radius of this circle is 11 centimeters. There’s also a chord 𝐵𝐶 and point 𝐴 lies somewhere along this chord such that the length of 𝐴𝐵 is three times the length of 𝐴𝐶. So, perhaps, point 𝐴 is here. We don’t know the length of 𝐴𝐵 and 𝐴𝐶, but we do know the ratio between these two lengths. If 𝐴𝐵 is three 𝐴𝐶, then if 𝐴𝐶 is 𝑥 centimeters for some nonzero value of 𝑥, 𝐴𝐵 will be three 𝑥 centimeters. We also know that point 𝐴 is eight centimeters away from point 𝑀, so we can add the length of this line segment to our diagram.
We’re asked to calculate the length of the chord 𝐵𝐶, so we need to determine the value of this unknown 𝑥. The information we’re 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 𝐴 be a point inside circle 𝑀 and let the line segment 𝐵𝐶 be a chord passing through 𝐴. Then the negative of 𝑃 sub 𝑀 of 𝐴 is equal to 𝐴𝐵 multiplied by 𝐴𝐶. The notation of 𝑃 sub 𝑀 of 𝐴 means the power of point 𝐴 with respect to circle 𝑀 and is defined to be equal to 𝐴𝑀 squared minus 𝑟 squared. That’s the square of the distance between points 𝐴 and 𝑀 minus the square of the radius.
We know both of these lengths. 𝐴𝑀 is equal to eight and 𝑟 is equal to 11. So, we can deduce that 𝑃 sub 𝑀 of 𝐴 is equal to eight squared minus 11 squared. That’s 64 minus 121 which is negative 57. We have then that negative negative 57, or simply 57, is equal to 𝐴𝐵 multiplied by 𝐴𝐶. Remember, we wrote down expressions for the lengths of 𝐴𝐵 and 𝐴𝐶 involving an unknown 𝑥. 𝐴𝐵 was defined to be three 𝑥 centimeters, and 𝐴𝐶 was defined to be 𝑥 centimeters. So, we have the equation 57 is equal to three 𝑥 multiplied by 𝑥, which simplifies to 57 is equal to three 𝑥 squared.
We can now solve this equation to determine the value of 𝑥. First, we divide both sides of the equation by three, giving 19 is equal to 𝑥 squared. Next, we find the square root of each side of this equation, taking only the positive value as we require 𝐴𝐵 and 𝐴𝐶 to be positive because they are lengths. So, we find that 𝑥 is equal to the square root of 19.
Finally, we need to calculate the length of 𝐵𝐶, which is equal to the length of 𝐴𝐵 plus the length of 𝐴𝐶, and that’s three 𝑥 plus 𝑥, or simply four 𝑥. We’ve just determined that 𝑥 is equal to the square root of 19, so we can substitute this value for 𝑥. And we find that 𝐵𝐶 is equal to four root 19. We’re asked to give our answer to the nearest hundredth, so we need to evaluate this as a decimal. It’s 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 𝐵𝐶 to the nearest hundredth is 17.44 centimeters.