$A$ ladder is placed leaning on a wall. Its upper end reaches to the height of $12 \, m$ on the wall and its lower end rests $9 \, m$ away from the base of the wall. Find the length of the ladder in meters.

In the figure,if $\triangle ABC \sim \triangle DEF$ and their side lengths (in $cm$) are as marked,find the lengths of the sides of each 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 $m$).

In $\Delta PQR$,$m \angle Q = 90^\circ$. If $PQ = 5$ and $PR = 13$,find the area of $\Delta PQR$.

$A$ flag pole $18 \, m$ high casts a shadow $9.6 \, m$ long. Find the distance of the top of the pole from the far end of the shadow. (in $m$)

Prove that in a parallelogram,the sum of squares of all the sides is equal to the sum of squares of the diagonals.