A series of infinite concentric squares are drawn as shown below. Starting with the first square $ABCD,$ subsequent squares drawn are $PQRS, XYZW$ and so on as shown in the diagram. If areas of the squares $ABCD, PQRS, XYZW,...$ are $1, \frac{3}{2}, \frac{7}{4}, \frac{15}{8}...$ and so on, then find the area of the diagram when the infinite number square is drawn.

Three regular hexagons are drawn such that their diagonals cut each other at the same point and area $A_1:A_2:A_3=1:2:3$. Then the ratio of the length of the sides of the regular hexagons (from the smallest to the largest) is:

Two circles $C_1$ and $C_2$ of radius $2$ and $3$ respectively touch each other as shown in the figure. If $AD$ and $BD$ are tangents then the length of $BD$ is

A right angled triangle is drawn on a plane such that sides adjacent to right angle are $3 cm$ and $4 cm$. Now three semi-circles are drawn taking all three sides of the triangle as diameters respectively (as shown in the figure). What is the area of the shaded regions $A_1+A_2$

In the given figure

$AB=20$

$BC=15$

$CA=19$

Calculate $a,b,c$