### Video Transcript

If π΄π΅πΆπ· is a rhombus, which line is the perpendicular bisector of line segment π΄πΆ? Option (A) line π΄π΅, option (B) line π΄π·, option (C) line πΆπ·, or option (D) line π΅π·.

Letβs begin by recalling what it means for a shape to be a rhombus. A rhombus is a quadrilateral or four-sided shape with all four sides equal in length. Weβre not told anything about the shape or size of this rhombus. But weβll know that as itβs a rhombus, the four sides will be the same length. So we could draw it like this or even like this.

When it comes to labeling the vertices of this rhombus, the ordering is important. We must go in order from π΄ to π΅ to πΆ to π·. But we can do this in either the clockwise or counterclockwise directions. The labeling on both of these rhombuses would be equally valid.

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. Weβre asked about the perpendicular bisector of this line. So weβll need to remember that perpendicular means at 90 degrees and the bisector will cut it exactly in half. So weβre looking for the line which cuts the line segment π΄πΆ exactly in half at 90 degrees.

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. The same is true for the other two lines. Neither the line π΅πΆ nor the line πΆπ· would be the perpendicular bisector of line segment π΄πΆ. In fact, the perpendicular bisector of line segment π΄πΆ would need to cut somewhere through this central section, looking something like this.

We should remember that an important property of rhombuses is that the diagonals of a rhombus are perpendicular bisectors. 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).

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. 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.