In $\Delta XYZ$,$P$ and $Q$ are the midpoints of $\overline{XY}$ and $\overline{XZ}$ respectively. If the area of $\Delta XYZ$ is $140$,find the area of $\Delta XPQ$.

$\Delta PQR \sim \Delta XYZ$ for the correspondence $PQR \leftrightarrow YZX$. If $m \angle P = 2 m \angle Q$ and $m \angle X = 120^\circ$,then $m \angle Y = \dots$ (in $^\circ$)

Hypotenuse of a right triangle is $25 \, cm$ and out of the remaining two sides,one is longer than the other by $5 \, cm$. Find the lengths of the other two sides.

In rhombus $ABCD$,$AC = 16$ and $BD = 30$. Find its perimeter.

In $\triangle ABC$,$AB = 24 \, cm$,$BC = 10 \, cm$,and $AC = 26 \, cm$. Is this triangle a right triangle? Give reasons for your answer.

In $\Delta ABC$,$D$ is the midpoint of $\overline{BC}$. $A$ line passing through $G$ (the centroid) intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N$. If $3 AN = 2 AC$,prove that $AM = 2 MB$.