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A circle with center π has a radius of 11 centimeters.
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Point π΄ lies eight centimeters from π and belongs to the chord π΅πΆ.
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Given that π΄π΅ equals three π΄πΆ, calculate the length of the line segment π΅πΆ, giving your answer to the nearest hundredth.
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Letβs begin by drawing a diagram to help.
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We have a circle with its center at π and the radius of this circle is 11 centimeters.
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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.
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So, perhaps, point π΄ is here.
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We donβt know the length of π΄π΅ and π΄πΆ, but we do know the ratio between these two lengths.
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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.
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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.
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Weβre asked to calculate the length of the chord π΅πΆ, so we need to determine the value of this unknown π₯.
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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.
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This states that let π΄ be a point inside circle π and let the line segment π΅πΆ be a chord passing through π΄.
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Then the negative of π sub π of π΄ is equal to π΄π΅ multiplied by π΄πΆ.
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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.
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Thatβs the square of the distance between points π΄ and π minus the square of the radius.
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We know both of these lengths.
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π΄π is equal to eight and π is equal to 11.
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So, we can deduce that π sub π of π΄ is equal to eight squared minus 11 squared.
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Thatβs 64 minus 121 which is negative 57.
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We have then that negative negative 57, or simply 57, is equal to π΄π΅ multiplied by π΄πΆ.
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Remember, we wrote down expressions for the lengths of π΄π΅ and π΄πΆ involving an unknown π₯.
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π΄π΅ was defined to be three π₯ centimeters, and π΄πΆ was defined to be π₯ centimeters.
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So, we have the equation 57 is equal to three π₯ multiplied by π₯, which simplifies to 57 is equal to three π₯ squared.
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We can now solve this equation to determine the value of π₯.
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First, we divide both sides of the equation by three, giving 19 is equal to π₯ squared.
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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.
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So, we find that π₯ is equal to the square root of 19.
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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₯.
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Weβve just determined that π₯ is equal to the square root of 19, so we can substitute this value for π₯.
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And we find that π΅πΆ is equal to four root 19.
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Weβre asked to give our answer to the nearest hundredth, so we need to evaluate this as a decimal.
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Itβs 17.4355 continuing, which to the nearest hundredth is 17.44.
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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.