In a $△ABC$, $AD$ is drawn perpendicular to $BC$. Prove that $A{B}^2-B{D}^2=A{C}^2-C{D}^2$.

In a right-angled triangle, prove that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Using the theorem, determine the length of $AD$ in terms of $b$ and $c$.

$ABC$ is a right angle triangle, right-angled at $C$. If $p$ is the length of the perpendicular from $C$ to $AB$ and $a, b, c$ have the usual meaning, then prove that $pc=ab$.

$ABC$ is a right angle triangle, right-angled at $C$. If $p$ is the length of the perpendicular from $C$ to $AB$ and $a, b, c$ have the usual meaning, then prove that $\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}$.

A $5 \mathrm{m}$ long ladder is placed leaning towards a vertical wall such that it reaches the wall at a point $4 \mathrm{m}$ high. If the foot of the ladder is moved $1.6 \mathrm{m}$ towards the wall, then find the distance by which the top of the ladder would slide upwards on the wall.

For going to a city $B$ from city $A$, there is a route via city $C$ such that $AC\perp CB$, $AC=2x \mathrm{km}$ and $CB=2\left(x+7\right) \mathrm{km}$. It is proposed to construct a $26 \mathrm{km}$ highway which directly connects the two cities $A$ and $B$. Find how much distance will be saved in reaching city $B$ from city $A$ after the construction of the highway.

In $△PQR$, $PD\perp QR$ such that $D$ lies on $QR$. If $PQ=a$, $PR=b$, $QD=c$ and $DR=d$, prove that $\left(a+b\right)\left(a-b\right)=\left(c+d\right)\left(c-d\right)$.

In a quadrilateral $ABCD$, $\angle A+\angle D=90°$. Prove that $A{C}^2+B{D}^2=A{D}^2+B{C}^2$.

In a right-angled triangle, the bisector of the right angle divides the hypotenuse in the ratio $m:n$. If in $\frac{m^k}{n^k}$ ratio the hypotenuse is divided by the altitude dropped from the vertex of the right angle then find $k$.