If the measures of the sides of a triangle are $\ldots \ldots \ldots,$ then it is not a right-angled triangle.

$A$ $5\, m$ long ladder is placed leaning against a vertical wall such that it reaches the wall at a point $4 \,m$ high. If the foot of the ladder is moved $1.6 \,m$ towards the wall,find the distance by which the top of the ladder would slide upwards on the wall (in $m$).

It is given that $\triangle ABC \sim \triangle DFE$,$\angle A = 30^{\circ}$,$\angle C = 50^{\circ}$,$AB = 5 \, cm$,$AC = 8 \, cm$ and $DF = 7.5 \, cm$. Then,the following is true:

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

In the figure,$O$ is a point in the interior of a triangle $ABC$,$OD \perp BC$,$OE \perp AC$,and $OF \perp AB$. Show that $OA^{2} + OB^{2} + OC^{2} - OD^{2} - OE^{2} - OF^{2} = AF^{2} + BD^{2} + CE^{2}$.

Find the altitude of an equilateral triangle of side $8 \, cm$. (in $cm$)