Radius of a circle with centre $O$ is $41$ units. Length of a chord $\mathrm{PQ}$ is $80$ units, if the distance of the chord from the centre of the circle is $k \mathrm{cm}$ then find $k$.

In figure given below, centre of two circles is $O$. Chord $\mathrm{AB}$ of bigger circle intersects the smaller circle in points $\mathrm{P}$ and $\mathrm{Q}$. Show that $\mathrm{AP}=\mathrm{BQ}$

Prove that, if a diameter of a circle bisects two chords of the circle then those two chords are parallel to each other.

Radius of circle is $10 \text{cm}$. There are two chords of length $16$ $\mathrm{cm}$ each. If the distance of these chords from the centre of the circle is $k \mathrm{cm}$ then find $k$.

In a circle with radius $13$$\text{ cm}$, two equal chords are at a distance of $5$$\text{ cm}$ from the centre. If the lengths of the chords is $k \mathrm{cm}$ then find $k$.

Seg $\mathrm{PM}$ and seg $\mathrm{PN}$ are congruent chords of a circle with center $C$. Show that the ray $\mathrm{PC}$ is the bisector of $\angle \mathrm{NPM}$

Construct $\Delta ABC$ such that $\angle B=100°, BC=6.4, \angle C=50°$ and construct its incircle.

Construct $\Delta PQR$ such that $\angle P=70°, \angle R=50°, QR=7.3$ $\text{cm}$, and construct its circumcircle.

Construct $\Delta XYZ$ such that $XY=6.7$$\text{cm}$, $YZ=5.8$$\text{cm}$, $XZ=6.9$$\text{cm}$. Construct its incircle.