The radius of a circle with center $O$ is $\sqrt{50} \mathrm{cm}$. $A$ and $C$ are two points on the circle, and $B$ is a point inside the circle. The length of $AB$ is $6 \mathrm{cm}$, and the length of $BC$ is $2 \mathrm{cm}$. The angle $ABC$ is a right angle. Find the square of the distance $OB$.

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: