Equilateral triangles are drawn on the three sides of a right-angled triangle. Show that the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides.

In the given figure,

$\frac{\mathrm{QT}}{\mathrm{PR}} =\frac{\mathrm{QR}}{\mathrm{QS}} \mathrm{and} \angle 1=\angle 2$

Prove that $△\mathrm{PQS}~△\mathrm{TQR}$

Ravi is $1.82 \mathrm{m}$ tall. He wants to find the height of a tree in his backyard. From the tree's base he walked $12.20 \mathrm{m}$ along the tree's shadow to a position where the end of his shadow exactly overlaps the end of the tree’s shadow. He is now $6.10 \mathrm{m}$ from the end of the shadow. How tall is the tree? Express your answer in metre(m).

$△\mathrm{ABC} \mathrm{and} △\mathrm{AMP}$ are two right triangles right angled at $\mathrm{B}$ and $\mathrm{M}$ respectively.

Prove that $△\mathrm{ABC}~△\mathrm{AMP}$

$△\mathrm{ABC} \mathrm{and} △\mathrm{AMP}$ are two right triangles right angled at $\mathrm{B}$ and $\mathrm{M}$ respectively.

Prove that $\frac{\mathrm{CA}}{\mathrm{PA}} =\frac{\mathrm{BC}}{\mathrm{MP}}$

In a right triangle $\mathrm{ABC}$ right angled at $\mathrm{C}$, $\mathrm{P} \mathrm{and} \mathrm{Q}$ are points on sides $\mathrm{AC} \mathrm{and} \mathrm{CB}$ respectively which divides these sides in the ratio of $2:1$

Prove that   $9{\mathrm{AQ}}^2=9{\mathrm{AC}}^2+4{\mathrm{BC}}^2$