Line segment BC has been bisected. So BD = CD, which is the same as answer choice 2.
NOTE that CD = DC, although it more common to name line segments alphabetically.
James S.
tutor
You contradicted yourself. You first said it wasn't possible, and then you said it could be done under certain conditions.
That is exactly what the problem is saying: that the perpendicular bisector of a triangle's side does go through the opposite vertex.
This only happens with isoceles triangles, and clearly it does not happen with other types of triangles.
Report
11/22/23
Michael W.
Well done James. Like you said, a perpendicular bisector can only be drawn in isosceles triangles. Knowing this triangle is isosceles, AB = AC would also be a valid answer if it were provided as an option.
Report
11/25/23
James S.
tutor
Agreed.
Report
11/25/23
James S.
tutor
The vertex of the triangle through which the perpendicular bisector was drawn is A, not B. As you mentioned, Michael, that means that AB=AC, not AB=BC.
Report
11/25/23
Doug C.
It is not possible to construct a perpendicular bisector of segment BC that is guaranteed to go through vertex A. It is possible to construct a line through A that is perpendicular to segment BC (altitude) or a line segment from A to the midpoint of BC (median). Only if AB = BC (triangle is isosceles) will the altitude and median be the same segment. So, if indeed he was able to construct a line through that is both perpendicular to BC and bisects BC, then AB must equal BC.11/22/23