M L Aggarwal Solutions for Exercise 1: EXERCISE 15.1

$ABC$ is an isosceles triangle inscribed in a circle. If  $AB= AC=12\sqrt{5} \mathrm{cm}$ and $BC= 24 \mathrm{cm}$. Find the radius of the circle.

In an equilateral triangle, prove that the centroid and the circumcentre of the triangle coincide.

In the figure given below, $OD$ is perpendicular to the chord $AB$ of a circle whose centre is $O$. If $BC$ is a diameter, show that $CA=2OD.$

In the figure given below, $O$ is the centre of a circle. If $AB$ and $AC$ are chords of the circle such that $AB=AC$ and$OP\perp AB, OQ\perp AC$, prove that $PB=QC$.

In the figure given below, a line $l$ intersects two concentric circles at the points $A,B,C \& D.$,Prove that $AB=CD.$.

In the figure $\left(2\right)$ given below, chords $AB$ and $CD$ of a circle with centre $O$ intersect at $E.$  If $OE$ bisects $\angle AED$, prove that $AB=CD.$

In the figure $\left(\mathrm{i}\right)$ given below, $AD$ is a diameter of a circle with centre $O.$ If $AB \parallel CD$ , prove that $AB=CD.$

In the figure $\left(\mathrm{ii}\right)$ given below, $AB$ and $CD$ are equal chords of a circle with $\left(\mathrm{i}\right) AE=CE$ $\left(\mathrm{ii}\right) BE=DE$