Question Video: Finding the Length of a Chord in a Circle Using the Given Relation between the Chords Mathematics

A circle with center 𝑀 has a radius of 11 cm. Point 𝐴 lies 8 cm from 𝑀 and belongs to the chord 𝐡𝐢. Given that 𝐴𝐡 = 3𝐴𝐢, calculate the length of 𝐡𝐢, giving your answer to the nearest hundredth.

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Video Transcript

A circle with center 𝑀 has a radius of 11 centimeters. Point 𝐴 lies eight centimeters from 𝑀 and belongs to the chord 𝐡𝐢. Given that 𝐴𝐡 equals three 𝐴𝐢, calculate the length of the line segment 𝐡𝐢, 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 𝐡𝐢 and point 𝐴 lies somewhere along this chord such that the length of 𝐴𝐡 is three times the length of 𝐴𝐢. So, perhaps, point 𝐴 is here. We don’t know the length of 𝐴𝐡 and 𝐴𝐢, but we do know the ratio between these two lengths. If 𝐴𝐡 is three 𝐴𝐢, then if 𝐴𝐢 is π‘₯ centimeters for some nonzero value of π‘₯, 𝐴𝐡 will be three π‘₯ 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 𝐡𝐢, so we need to determine the value of this unknown π‘₯. 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 𝐡𝐢 be a chord passing through 𝐴. Then the negative of 𝑃 sub 𝑀 of 𝐴 is equal to 𝐴𝐡 multiplied by 𝐴𝐢. 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 π‘Ÿ 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 π‘Ÿ 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 𝐴𝐡 multiplied by 𝐴𝐢. Remember, we wrote down expressions for the lengths of 𝐴𝐡 and 𝐴𝐢 involving an unknown π‘₯. 𝐴𝐡 was defined to be three π‘₯ centimeters, and 𝐴𝐢 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 𝐴𝐡 and 𝐴𝐢 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 𝐡𝐢, which is equal to the length of 𝐴𝐡 plus the length of 𝐴𝐢, and that’s three π‘₯ plus π‘₯, or simply four π‘₯. We’ve just determined that π‘₯ is equal to the square root of 19, so we can substitute this value for π‘₯. And we find that 𝐡𝐢 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 𝐡𝐢 to the nearest hundredth is 17.44 centimeters.

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