In $\Delta ABC$,$\overline{AD}$ is a median and $AB^2 + AC^2 = 148$. If $AD = 7$,then $BC = \ldots$

$A$ $17 \, m$ long ladder leans on a wall such that its lower end rests $8 \, m$ away from the base of the wall. Then,its upper end reaches to the height of $\ldots \ldots \ldots \, m$ on the wall.

$\Delta XYZ \sim \Delta DEF$ for the correspondence $XYZ \leftrightarrow EDF$. If $XY = 3, YZ = 4, ZX = 6$ and $DF = 12$,find the perimeter of $\Delta DEF$.

In $\square ABCD$,$P \in \overline{AB}$,$Q \in \overline{BC}$,$R \in \overline{CD}$,and $S \in \overline{DA}$ such that $\frac{AP}{PB} = \frac{AS}{SD}$ and $\frac{CB}{QB} = \frac{CR}{RD}$. Prove that $\overline{PS} \parallel \overline{QR}$.

Is it true to say that if in two triangles,an angle of one triangle is equal to an angle of another triangle and two sides of one triangle are proportional to the two sides of the other triangle,then the triangles are similar? Give reasons for your answer.

If $\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$,and given $AB + BC = 12$,$PQ + QR = 15$,and $AC = 8$,find the length of $PR$.