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 $\Delta RPQ, \angle {120}^{\mathrm{o}}$ angle bisector of $\angle P$ meets $RQ$ at $S$. $PQ=9 cm$, $PS=6 cm$. Then the value of $PR$ (in $cm$) is:

In the diagram given below in $\Delta ABC$ in which $AB=5 cm, BC=12 cm, AC=13 cm$. Side $AB$ and $BC$ are divided in $n$-equal parts by $n-1$ equally spaced points as shown in the diagram. $A_1$ joined to $B_1, A_2$ joined to $B_2$ and so on, then answer the following questions:

The value of '$n$' for which $A_{n-1}{B}_{n-1}+A_{n-2}{B}_{n-2}+A_{n-3}{B}_{n-3}+\mathrm{...}+AC=130 cm$ is ____________

In the diagram given below in $\Delta ABC$ in which $AB=5 cm, BC=12 cm, AC=13 cm$. Side $AB$ and $BC$ are divided in $n$-equal parts by $n-1$ equally spaced points as shown in the diagram. $A_1$ joined to $B_1, A_2$ joined to $B_2$ and so on, then answer the following questions:

What will be the value of $n$ for which:

Area of $\Delta ABC+$ Area of $\Delta A_{1}B{B}_1+$ Area of $\Delta A_{2}B{B}_2+$ Area of $\Delta A_{3}B{B}_3+\mathrm{...}+$ Area of $\Delta A_{n-1}B{B}_{\left(n-1\right)}=66 c{m}^2$

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: