Challenges on Chords and Distance from the Center

Q. In the figure, $AB$ and $PQ$ are chords of a circle with center $O.$ Point $M$ lies on $AB$ such that $OM$ is perpendicular to $AB,$ and point $N$ lies on $PQ$ such that $ON$ is perpendicular to $PQ.$ Find the length of $PQ.$

1. $4$ cm
2. $8$ cm
3. $10$ cm
4. $5$ cm

Q. Find the angle subtended by a chord of length 12 units to the center of a circle having a radius of √72 units.

degrees

Q. A square is inscribed in a circle of area $2\pi {\text{unit}}^2,$ as shown in figure. Another circle is inscribed in the square. Find the area of the shaded region.

1. $(4-\pi ){\text{unit}}^2$
2. $(2-\pi ){\text{unit}}^2$

Q. Given that $\angle YPZ={110}^o$. The angle ∠XZY is degrees, where P is the center of the given circle.

Q. In the given figure, $O$ is the center of the circle. $OD=4\text{units}$ and $DC=10\text{units}.$ $OD$ bisects the cord $AB.$ If area of $△AOD$ is $10{\text{unit}}^2$, find the area of the quadrilateral $AOBC.$

1. $70{\text{units}}^2$
2. $35{\text{units}}^2$