If 𝐴𝐵𝐶𝐷 is a rhombus, which line is the perpendicular bisector of line segment 𝐴𝐶? Option (A) line 𝐴𝐵, option (B) line 𝐴𝐷, option (C) line 𝐶𝐷, or option (D) line 𝐵𝐷.
Let’s 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. We’re not told anything about the shape or size of this rhombus. But we’ll know that as it’s 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 𝐴 to 𝐵 to 𝐶 to 𝐷. 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 we’ve correctly followed the labeling convention, when we draw in the line segment 𝐴𝐶, we can see that it is in fact one of the diagonals of the rhombus. We’re asked about the perpendicular bisector of this line. So we’ll need to remember that perpendicular means at 90 degrees and the bisector will cut it exactly in half. So we’re looking for the line which cuts the line segment 𝐴𝐶 exactly in half at 90 degrees.
If we look at the rhombuses, we can see that neither the line 𝐴𝐵 nor the line 𝐴𝐷 is the perpendicular bisector of the line segment 𝐴𝐶. The same is true for the other two lines. Neither the line 𝐵𝐶 nor the line 𝐶𝐷 would be the perpendicular bisector of line segment 𝐴𝐶. In fact, the perpendicular bisector of line segment 𝐴𝐶 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 𝐵𝐷, which is the answer given in option (D).
Notice that we can use the line notation rather than just the line segment as it’s the whole line 𝐵𝐷 that’s the perpendicular bisector of line segment 𝐴𝐶. 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 𝐵𝐷 as the perpendicular bisector of line segment 𝐴𝐶.