Question

Geometry Question: If the median to a side is a triangle is also an altitude to that side, then the triangle is isosceles. (#8 in picture)

Answer 1

8.

A median of a triangle is a line segment drawn from a vertex to the midpoint of the opposite side of the vertex. So:

$\begin{gathered} AM\cong MB \\ MC\cong CM_{\text{ }}Reflexive_{\text{ }}property \\ m\angle AMC\cong m\angle BMC_{\text{ }}Perpendicular_{\text{ }}bisector \end{gathered}$

Therefore:

$\Delta AMC\cong\Delta BMC_{\text{ }}SAS_{\text{ }}Theorem$$m\angle CAM\cong\angle CBM_{\text{ }}CPCTC$

We can conclude that:

$\Delta ABC$

Is an isosceles triangle