In the figure given below $∆PQR$ is a right angled triangle with $\angle P=90°$. The center of the incircle of the given triangle is $O_2$. Circles with centers $O_1$ and $O_2$ touch the circle and two sides as shown in the figure. If the radius of the incircie of $∆PQR$ is $1$ cm and $BR : BQ = 2:3$, then find the value of  $r_1: r_2$ (where $r_1$ is the radius of circle with center $O_2$ and $r_2$ is the radius of circle with center $O_2$)

In the given figure $PQ$ and $PR$ are tangents of a semicircle. Center $\text{'}O\text{'}$ of this semicircle lies on $QR$. If $QO=2 \mathrm{cm}$ and $OR=4 \mathrm{cm}$. If $\angle P={90}^{\circ }$ the radius of the semicircle is:

Triangle $ABC$ is a right angled triangle. $D$ and $E$ are mid points of $AB$ and $BC$ respectively. Read the following statements.

I. $AE=19$

II. $CD=22$,

III. Angle $B$ is a right angle.

Which of the following statements would be sufficient to determine the length of $AC$?

In the Given figure $∆ABC$ is an equilateral triangle $BP\perp AC$ and $PR\perp AB, PQ\perp BC$. The ratio of area of $∆PRB$ and $∆PQC$ is: