Question

1. in △abc, (overline{bd}) is the perpendicular bisector of (overline{ac}). based upon this information, which statements can be proven? i. (overline{bd}) is a median. ii. (overline{bd}) bisects ∠abc. iii. △abc is isosceles. iv. △abc is a right triangle.

Explanation:

Step1: Recall median definition

A median of a triangle is a line - segment joining a vertex to the mid - point of the opposite side. Since $\overline{BD}$ is the perpendicular bisector of $\overline{AC}$, it divides $\overline{AC}$ into two equal parts ($AD = DC$), so $\overline{BD}$ is a median.

Step2: Analyze angle - bisector condition

Just because $\overline{BD}$ is the perpendicular bisector of $\overline{AC}$, we cannot say that it bisects $\angle ABC$ without additional information. There is no given condition to prove $\angle ABD=\angle DBC$.

Step3: Check isosceles triangle property

Since $\overline{BD}$ is the perpendicular bisector of $\overline{AC}$, we can use the property of the perpendicular bisector theorem. Points on the perpendicular bisector of a line - segment are equidistant from the endpoints of the line - segment. So, $AB = BC$, and $\triangle ABC$ is isosceles.

Step4: Examine right - triangle condition

There is no information given to suggest that any of the angles in $\triangle ABC$ is a right angle.

Answer:

I. $\overline{BD}$ is a median; III. $\triangle ABC$ is isosceles.