Level of Difficulty

Find the length of a chord that is at a distance of $12 \mathrm{cm}$ from the centre of a circle of radius $13 \mathrm{cm}$.
 

If $\mathrm{O}$is the center of circle, find $\angle \mathrm{x}$.

The sum of the interior angles of a regular polygon is $40$ times the exterior angle. Find the number of sides of the polygon.

An arc $\mathrm{AB}$of a circle subtends an angle $\text{'}\mathrm{x}\text{'}$radian at the center of the circle. If the area of the sector $\mathrm{AOB}$is equal to the square of the length of the arc $\mathrm{AB}$, then $\mathrm{x}$is:

In a circle, the height of an arc is $21 \mathrm{cm}$ and the diameter is $84 \mathrm{cm}$. Find the chord of 'half of the are?

In KyaKya Island, there is a circular park. There are four points of entry into the park, namely –$\mathrm{P}. \mathrm{Q}. \mathrm{R} \mathrm{and} \mathrm{S}.$ The king of the island His excellency Mr. Honolulu got three paths constructed which connected the points $\mathrm{PQ}, \mathrm{RS}, \mathrm{and} \mathrm{PS}$. The length of the path $\mathrm{PQ}$is $10 \mathrm{units}$, and the length of the path RSIs $7 \mathrm{units}.$ Later, the municipal corporation extended the paths $\mathrm{PQ}$and $\mathrm{RS}$past $\mathrm{Q}$and $\mathrm{R}$respectively, and they meet at a point $\mathrm{T}$on the main road outside the park. The path from $\mathrm{Q}$to $\mathrm{T}$measures $8$units, and it was found that the $\mathrm{ZPTS}$ is $60°.$ Find the area (in square units) enclosed by the paths $\mathrm{PT}, \mathrm{TS}, \mathrm{and} \mathrm{PS}.$

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In quadrilateral $\mathrm{PQRS}, \mathrm{PQ} = 5\mathrm{unit}$ , $\mathrm{QR} = 17$ units, $\mathrm{RS} = 5 \mathrm{units}$, and $\mathrm{PS} 9 \mathrm{units}.$ The length of the Diagonal $\mathrm{QS}$can be:

In a square $\mathrm{PORS}$, $\mathrm{A}$and $\mathrm{B}$are two points on $\mathrm{PS}$and $\mathrm{SR}$such that$\mathrm{PA} = 2\mathrm{AS}$ , and $\mathrm{RB} = 2\mathrm{BS}$ If $\mathrm{PQ}= 6$, the area of the triangle $\mathrm{ABQ}$Is