B Nirmala Shastry Solutions for Exercise 1: EXERCISE 11

In $△\mathrm{ABC}, \mathrm{AB}=8$, find a and b if $\mathrm{a}+\mathrm{b}=32$.

In quadrilateral $\mathrm{ABCD}, \angle \mathrm{B}=\angle \mathrm{D}={90}^{°}.$ Prove that $2{\mathrm{AC}}^2-{\mathrm{BC}}^2={\mathrm{AB}}^2+{\mathrm{AD}}^2+{\mathrm{DC}}^2$.

In triangle $\mathrm{ABC}, \mathrm{AD}\perp \mathrm{BC}$, prove that $\left(\mathrm{AB}+\mathrm{AC}\right)\left(\mathrm{AC}-\mathrm{AB}\right)=\left(\mathrm{CD}+\mathrm{BD}\right)\left(\mathrm{CD}-\mathrm{BD}\right)$.

In $△\mathrm{ABC}$, $\angle \mathrm{B}=90°$ and $\mathrm{D}$ is the mid point of $\mathrm{BC}$, Prove that: ${\mathrm{AC}}^2={\mathrm{AD}}^2+3{\mathrm{CD}}^2$

In $△\mathrm{PQR}, \mathrm{PS}$ is perpendicular to $\mathrm{RQ}$ extended. Prove that ${\mathrm{PR}}^2={\mathrm{PQ}}^2+{\mathrm{QR}}^2+2\mathrm{QR}.\mathrm{SQ}$.

$\mathrm{P}$ is any point inside the rectangle $\mathrm{ABCD}.$Prove that; ${\mathrm{PA}}^2+{\mathrm{PC}}^2={\mathrm{PB}}^2+{\mathrm{PD}}^2$.

In the rectangle $ABCD, AB=12 cm, BC=7 cm. P \& R$ are mid-points of $AB \& CD. AS=BQ=2.5 cm.$  Find the perimeter of $PQRS$.

The upper part of a tree is broken by the wind and falls on the ground without being detached. If the top of the broken part touches the ground at a distance of $12$ feet from the foot of the tree, calculate the height at which it is broken if the original height is $24$ feet.