Application of Pythagoras Theorem

Define Apollonius theorem

In $△ABC$, if $M$ is the midpoint of side $BC$, then $A{B}^2+A{C}^2=2A{M}^2+2B{M}^2$.

$△ABC$ is an equilateral triangle. Point $D$ is on side $BC$ such that $BD=\frac{1}{5}BC$ then prove $25A{D}^2=21A{B}^2$

In a $△ABC, AB=AC=x, BC=10 cm$ and the area of $△ABC is 60 c{m}^2. Find x.$

Find the diagonal (in $\mathrm{cm}$) of a square whose side is $10 \mathrm{cm}$. $\left(\sqrt{2}=1.414\right)$

In $△ABC$, $AD$ is a median such that $\angle ADB$ is obtuse, then prove that $A{B}^2+ A{C}^2 = 2A{D}^2 + \frac{1}{2}B{C}^2$

 $AE=DC=13 \mathrm{cm}, BE=5 \mathrm{cm}, \angle ABC=90° \mathrm{and} AD=EC= x \mathrm{cm}.$ Calculate the length of $AB$ and the value of $x$.

The sides of a right angled triangle containing the right angle are $5x cm$  and $\left(3x-1\right) cm .$ If area of the triangle is $60 c{m}^2,$ calculate the lengths of the sides of the triangle.