R Chandra Solutions for Chapter: Circles, Exercise 1: TEST YOUR SKILLS

Prove that equal chords of a circle (or congruent circle) are equidistant from the centre.

In the figure $\mathrm{O}\mathrm{D}$ is perpendicular to the chord $\mathrm{A}\mathrm{B}$ of a circle with centre $\mathrm{O}$. If $\mathrm{B}\mathrm{C}$ is a diameter show that $\mathrm{A}\mathrm{C}\parallel \mathrm{O}\mathrm{D}$ and $\mathrm{A}\mathrm{C}=2\mathrm{O}\mathrm{D}$.

If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, prove that the chords are equal.

Show that the angles in the same segment of a circle are equal.

Show that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment then show that the four points are concyclic.

In the figure $\mathrm{A}\mathrm{B}$ is a diameter of the circle $\mathrm{C}\mathrm{D}$ is a chord equal to radius of the circle. $\mathrm{A}\mathrm{C}\mathrm{}$and $\mathrm{B}\mathrm{D}$ when produced meet at $\mathrm{E}$. Prove that $\mathrm{\angle }\mathrm{A}\mathrm{E}\mathrm{B}=60°$.

In the figure, $\mathrm{ABCD}$ is a cyclic quadrilateral such that $\mathrm{AC} \mathrm{and} \mathrm{BD}$ are its diagonals. If $\angle \mathrm{DBC}=55° \mathrm{and} \angle \mathrm{BAC}=45°$ and $\angle \mathrm{BCD}=k°$, then find $k$.