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In the shown figure, π΄π equals 7.5 centimetres, π΅π equals 14.5 centimetres, and π΄πΆ equals 20 centimetres.
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Given that all sides of triangle π΄π΅πΆ are tangent to the shown circle, determine the length of π΅πΆ.
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Letβs begin by adding the length that weβre given in the question to the diagram.
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So we have the lengths of two of the sides of this triangle and weβre looking to determine the length of π΅πΆ, which is the third side.
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Weβre also given the key piece of information that all three sides of the triangle π΄π΅πΆ are tangent to the shown circle.
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Letβs think about what we know about the lengths of tangents drawn from exterior points to circles.
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Hereβs an important fact.
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If two segments from the same exterior point are tangent to a circle, then they are congruent.
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In practice, what this means is that the two line segments drawn from the point π΄ to the circle are both equal in length.
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The same is true for the segments drawn from π΅ and those drawn from πΆ.
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How would this help us with answering the question?
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Well, remember we want to determine the length of π΅πΆ.
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So in our diagram, we need to know the length of the pink segment and the length of the green segment.
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Using the result we just discussed, we actually already know the length of the pink segment.
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Itβs equal to π΅π which is 14.5 centimetres.
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So we need to think about how weβre going to calculate the length of the green segment.
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And to do this, Iβm going to add in a couple of labels on the other two sides of this triangle.
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So just as we have the point π on the side π΄π΅, we now have the point π on the side π΄πΆ and the point π on the side π΅πΆ, which are the points where these tangents touch the circle.
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We know the full length of the side π΄πΆ.
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But we want to know how much of this is due to the orange part π΄π and how much is due to the green part ππΆ.
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Well, applying the same result again, we know that the line segment π΄π is congruent to π΄π.
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And therefore, itβs equal to 7.5 centimetres.
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The line segment πΆπ can therefore be found by subtracting π΄π from the length of π΄πΆ: 20 minus 7.5.
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So we know that πΆπ is 12.5 centimetres.
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Applying our key result a third time, we know that the two segments drawn from the point πΆ are congruent to each other.
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And therefore, πΆπ is congruent to πΆπ.
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πΆπ is 12.5 centimetres.
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So our final step in this problem, we need to determine the length of π΅πΆ by summing the two segments π΅π and πΆπ.
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So π΅πΆ is equal to 14.5 plus 12.5.
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Itβs 27 centimetres.
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The key result which we applied three times within this question is that if two segments from the same exterior point are tangent to a circle, then they are congruent β meaning theyβre equal in length.