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Given that π΄π΅ is equal to πΆπ·, ππΆ is equal to 10 centimeters, and π·πΉ is equal to eight centimeters, find the length of line segment ππΈ.
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In this question, we are trying to calculate the length of ππΈ, which is the distance from the chord π΄π΅ to the center of the circle π.
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We begin by recalling that two chords of equal lengths are equidistant from the center.
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And in this question, we are told that the two chords π΄π΅ and πΆπ· are equal in length.
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This means that the length ππΉ must be equal to ππΈ.
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The line segment ππΈ perpendicularly bisects the chord π΄π΅.
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Likewise, ππΉ is the perpendicular bisector of πΆπ·.
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Since we are told π·πΉ is equal to eight centimeters, πΆπΉ, π΄πΈ, and π΅πΈ are all also equal to eight centimeters.
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Our next step is to consider the right triangle ππΉπΆ.
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Using the Pythagorean theorem, ππΉ squared plus πΆπΉ squared is equal to ππΆ squared.
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By subtracting πΆπΉ squared from both sides and substituting in the values of πΆπΉ and ππΆ, we have ππΉ squared is equal to 10 squared minus eight squared.
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This is equal to 36.
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Square rooting both sides of this equation and knowing that ππΉ must be a positive answer, we have ππΉ is equal to six.
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Since ππΉ is equal to six centimeters, ππΈ must also be equal to six centimeters.
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The perpendicular distance from the center of the chord π΄π΅ to the center of the circle π, which is the line segment ππΈ, is equal to six centimeters.