In a right angled triangle $\mathrm{ABC},\angle \mathrm{B}$ is right angle, side $\mathrm{AB}$ is half of the hypotenuse. $\mathrm{AE}$ is parallel to median $\mathrm{BD}$ and $\mathrm{CE}$ is parallel to $\mathrm{BA}$. What is the ratio of length of $\mathrm{BC}$ to that of $\mathrm{EC} ?$

$\mathrm{ABCD}$ is a quadrilateral in which $\frac{\mathrm{z}}{\mathrm{y}}=\frac{\mathrm{y}}{\mathrm{x}}=\frac{\mathrm{x}}{\mathrm{w}}=\mathrm{r}$ and $\mathrm{k}$ is an integer. Also $\{\mathrm{w},\mathrm{x}\}<90$ and $\{\mathrm{y},\mathrm{z}\}>90$ then the difference between the greatest angle and smallest angle (i.e., $\mathrm{z}-\mathrm{w}$ ) is

In the given figure $\mathrm{ABCD}$ is a cyclic quadrilateral, $\mathrm{DO}=8 \mathrm{cm}$ and $\mathrm{CO}=4 \mathrm{cm}$. $\mathrm{AC}$ is the angle bisector of $\angle \mathrm{BAD}$. The length of $\mathrm{AD}$ is equal to the length of $\mathrm{AB}. \mathrm{DB}$ intersects diagonal $\mathrm{AC}$ at $\mathrm{O}$, what is the length of the diagonal $\mathrm{AC} ?$

In the given diagram CT is tangent at C, making an angle of $\frac{\pi }{4}$ with CD. O is the centre of the circle. CD$=10$ cm. What is the perimeter of the shaded region ($\mathrm{\Delta }$ $\mathrm{AOC}$) approximately? 

If ${\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2=\mathrm{xy}+\mathrm{yz}+\mathrm{zx},$ then the triangle is :

In the adjoining figure there are two congruent regular hexagons each with side $6 \mathrm{cm}.$ 

What is the ratio of area of $∆\mathrm{BDF}$ and $∆\mathrm{PQR},$ if $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ are the mid-points of side $\mathrm{AF}, \mathrm{BC}$ and $\mathrm{DE} ?$

$\mathrm{ABCD}$ is a square, in which a circle is inscribed touching all the sides of a square. In the four corners of square $4$ smaller circles of equal radii are drawn, containing the maximum possible area. What is the ratio of the area of larger circle to that of sum of the areas of four smaller circles?

In the adjoining figure $\angle \mathrm{ACE}$ is a right angle. There are three circles which just touch each other where $\mathrm{AC}$ and $\mathrm{EC}$ are the tangents to all the three circles. What is the ratio of radii of the largest circle to that of the smallest circle?