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Given that π΄π΅ equals 24 centimeters, π΄π· equals 36 centimeters, π΄πΆ equals 18 centimeters, and πΈπ equals 15 centimeters, find the length of line segment π΄πΈ and line segment π·π.
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Starting by filling in the lengths on the diagram, we can also see that we have three parallel lines marked on this diagram.
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And we also have two transversals.
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A transversal is a line which passes through two lines in the same plane at two distinct points.
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We can use the fact that if three or more parallel lines are cut by two transversals, then they divide the transversals proportionally.
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As weβre given the lengths on the line π·π΅, we can use this to work out the proportional relationship between the segments.
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We can write that the segment π·π΄ over π΄π΅ is equal to the segment π΄πΈ over π΄πΆ.
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And we can then substitute the values that weβre given for each segment.
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On the left-hand side, we have the values of 36 over 24, which is equal to π΄πΈ over 18.
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To find the missing value for π΄πΈ, we will take the cross product.
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Noticing that 12 is a factor of both 36 and 24 means that we can simplify the fraction on the left-hand side as three over two.
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And thatβs equal to π΄πΈ over 18.
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Taking the cross product then, we have three times 18 is equal to two times π΄πΈ.
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So 54 equals two times π΄πΈ.
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Dividing both sides by two will give us that 27 equals π΄πΈ.
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And therefore, π΄πΈ is equal to 27 centimeters.
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And we found our first missing length.
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To find the next missing length of π·π, we can notice that, on the other transversal, the length πΈπ will be the corresponding length.
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Defining the length π·π as π, we can then write that π over 24 is equal to 15 over 18.
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We can simplify the fraction 15 over 18 to give us π over 24 equals five-sixths.
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We can then take the cross product.
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So π times six or six π is equal to 24 times five.
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And simplifying, we have six π equals 120.
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So π equals 20.
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And given that we define the length π·π as π, then we have π·π equals 20 centimeters.
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So our final answer is π΄πΈ equals 27 centimeters.
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π·π equals 20 centimeters.