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If π΄π΅πΆπ· is a rhombus, which line is the perpendicular bisector of line segment π΄πΆ?
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Option (A) line π΄π΅, option (B) line π΄π·, option (C) line πΆπ·, or option (D) line π΅π·.
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Letβs begin by recalling what it means for a shape to be a rhombus.
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A rhombus is a quadrilateral or four-sided shape with all four sides equal in length.
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Weβre not told anything about the shape or size of this rhombus.
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But weβll know that as itβs a rhombus, the four sides will be the same length.
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So we could draw it like this or even like this.
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When it comes to labeling the vertices of this rhombus, the ordering is important.
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We must go in order from π΄ to π΅ to πΆ to π·.
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But we can do this in either the clockwise or counterclockwise directions.
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The labeling on both of these rhombuses would be equally valid.
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So if weβve correctly followed the labeling convention, when we draw in the line segment π΄πΆ, we can see that it is in fact one of the diagonals of the rhombus.
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Weβre asked about the perpendicular bisector of this line.
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So weβll need to remember that perpendicular means at 90 degrees and the bisector will cut it exactly in half.
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So weβre looking for the line which cuts the line segment π΄πΆ exactly in half at 90 degrees.
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If we look at the rhombuses, we can see that neither the line π΄π΅ nor the line π΄π· is the perpendicular bisector of the line segment π΄πΆ.
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The same is true for the other two lines.
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Neither the line π΅πΆ nor the line πΆπ· would be the perpendicular bisector of line segment π΄πΆ.
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In fact, the perpendicular bisector of line segment π΄πΆ would need to cut somewhere through this central section, looking something like this.
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We should remember that an important property of rhombuses is that the diagonals of a rhombus are perpendicular bisectors.
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Therefore, we can give our answer that it would be the other diagonal, the line π΅π·, which is the answer given in option (D).
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Notice that we can use the line notation rather than just the line segment as itβs the whole line π΅π· thatβs the perpendicular bisector of line segment π΄πΆ.
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Even in this second example of a rhombus, which is a different size and the letters are in different positions, we still have the line π΅π· as the perpendicular bisector of line segment π΄πΆ.