In $\Delta ABC$,$m \angle A = m \angle B + m \angle C$. If $AB = 7$ and $BC = 25$,the perimeter of $\Delta ABC = ........$

In $\Delta ABC$,$m \angle B = 45^{\circ}$ and $\overline{AM}$ is an altitude,$M \in \overline{BC}$. If $BC = 7$ and $AM = 4$,then $AC = \ldots$

In $\Delta ABC$,$D$,$E$ and $F$ are the midpoints of $\overline{BC}$,$\overline{CA}$ and $\overline{AB}$ respectively. Then,$\text{Area}(BDEF) = \ldots \times \text{Area}(\Delta ABC)$.

Prove that the area of the equilateral triangle drawn on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the equilateral triangles drawn on the other two sides of the triangle.

As shown in the adjoining diagram of a staircase,the width of each step is $20 \, cm$ and the height is $15 \, cm$. Find the distance between points $A$ and $B$ in meters.

In $\Delta ABC$,$AB + BC = 23$,$BC + AC = 32$,and $AB + AC = 25$. Show that $\Delta ABC$ is a right-angled triangle.