Mizoram Board Solutions for Chapter: Circles, Exercise 6: EXERCISE (Optional)*

Prove that line of centres of two intersecting circles subtends equal angles at the two points of intersection.

Two chords $AB$ and $CD$ of lengths $5 cm$ and $11 cm$ respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between $AB$ and $CD$ is $6 cm$, find the radius of the circle.

The lengths of two parallel chords of a circle are $6 \mathrm{cm}$ and $8 \mathrm{cm}$. If the smaller chord is at distance $4 \mathrm{cm}$ from the centre, what is the distance of the other chord from the centre ?

Let the vertex of an angle $ABC$ be located outside a circle and let the sides of the angle intersect equal chords $AD$ and $CE$ with the circle. Prove that $\angle ABC$ is equal to half the difference of the angles subtended by the chords $AC$ and $DE$ at the centre.

Prove that the circle drawn with any side of a rhombus as diameter passes through the point of intersection of its diagonals.

Bisectors of angles $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$of a $∆\mathrm{ABC}$intersect its circumcircle at $\mathrm{D}, \mathrm{E}$ and $\mathrm{F}$, respectively. Prove that the angles of the $∆\mathrm{DEF}$are $90°-\frac{1}{2}\mathrm{A}, 90°-\frac{1}{2}\mathrm{B}$ and $90°-\frac{1}{2}\mathrm{C}$.

Two congruent circles intersect each other at points $\mathrm{A}$and $\mathrm{B}$. Through $\mathrm{A}$ any line segment $\mathrm{PAQ}$is drawn so that $\mathrm{P}$, $\mathrm{Q}$ lie on the two circles. Prove that $\mathrm{BP}=\mathrm{BQ}$.

In any $∆\mathrm{ABC}$, if the angle bisector of $\angle \mathrm{A}$ and perpendicular bisector of $\mathrm{BC}$intersect, prove that they intersect on the circumcircle of the $∆\mathrm{ABC}$.