The perimeter of an equilateral triangle is $18$. Then,the length of its altitude is $\ldots \ldots \ldots$

If in the figure, $O$ is the point of intersection of two chords $AB$ and $CD$ such that $OB = OD$, then triangles $OAC$ and $ODB$ are

In $\Delta XYZ$,$\overline{XM}$ is a median. If $XY = 11$ and $XM = YM = 30.5$,find $XZ$.

In $\Delta ABC$,$P, Q$ and $R$ are the midpoints of $\overline{BC}, \overline{CA}$ and $\overline{AB}$ respectively. Then,$Area(BCQR) = \ldots \ldots \ldots \times Area(ABC)$.

The two triangles in the figure are congruent using a congruence theorem. It is given that $OQ = OR$. Which of these conditions,along with the given condition,is sufficient to prove that the two triangles are congruent to each other?

As shown in the adjoining diagram,$\angle B$ is an obtuse angle and $\overline{AM}$ is an altitude in $\Delta ABC$. Prove that $AC^{2} = AB^{2} + BC^{2} + 2 \cdot BC \cdot BM$.