In $\Delta ABC$,the bisector of $\angle B$ intersects $\overline{AC}$ at $D$. If $\frac{AD}{DC} = \frac{3}{4}$ and $AB = 7.5$,then $BC = \ldots$

In $\Delta PQR$,$PQ = 25$ and $QR = 24$. If $PR = \dots$,then $\angle R$ is a right angle.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{AD}$ is a median. Prove that $AC^{2} = AD^{2} + 3BD^{2}$.

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.

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.

In a triangle,if the bisector of an angle bisects the side opposite to that angle,then prove that the triangle is an isosceles triangle.