In $\Delta PQR$,$\overline{PM}$ is a median. If $PQ^{2} + PR^{2} = 148$ and $PM = 7$,find $QR$.

Foot of a $10\,m$ long ladder leaning against a vertical wall is $6\,m$ away from the base of the wall. Find the height of the point on the wall where the top of the ladder reaches. (in $m$)

In rhombus $ABCD$,$\overline{AC} \cap \overline{BD} = \{O\}$. Then the area of $\Delta OAB = \ldots \ldots \ldots \ldots \times$ area of $\square ABCD$.

Out of the following triplets,$\ldots \ldots \ldots \ldots$ is not a Pythagorean triplet.

In $\Delta ABC,$ the midpoints of the sides are $P, Q, R.$ In $\Delta PQR,$ the midpoints of the sides are $X, Y, Z.$ If the area of $\Delta XYZ$ is $20,$ find the area of $\Delta PQR$ and $\Delta ABC.$

In a rectangle $ABCD$,if $AB + BC = 47$ and the diagonal $AC = 37$,find the lengths of $AB$ and $BC$,given that $AB > BC$.