M L Aggarwal Solutions for Chapter: Circles, Exercise 4: Chapter Test

In the adjoining figure, $TP \text{and} TQ$ are tangents at points $P \text{and} Q$ respectively to a circle with centre $O$. $R$ is a point on the circle. If $\angle PTQ=50°$, find $\angle OPQ$

In the figure given below, two circles with centres $\mathrm{A}$ and $\mathrm{B}$ touch externally. $\mathrm{PM}$ is a tangent to the circle with centre $\mathrm{A}$ and $\mathrm{QN}$ is a tangent to the circle with centre $\mathrm{B}$. If $\mathrm{PM}=15 \mathrm{cm},\mathrm{QN}=12 \mathrm{cm},\mathrm{PA}=17 \mathrm{cm}$ and $\mathrm{QB}=13 \mathrm{cm},$ then find the distance between
the centres $\mathrm{A}$ and $\mathrm{B}$ of the circles.

In figure, $\mathrm{CP}$ and $\mathrm{CQ}$ are tangents to a circle with centre $\mathrm{O}$. $\mathrm{ARB}$ is another tangent touching the circle at $\mathrm{R}$. If $\mathrm{C}\mathrm{P}=11 \mathrm{c}\mathrm{m}$ and $\mathrm{B}\mathrm{C}=7 \mathrm{c}\mathrm{m},$ then find the length of $\mathrm{BR}$.

In the adjoining figure, AB and CD are common tangents to two circles with centres O and O′ of equal radii. Prove that AB = CD.

$AB$ is a diameter and $AC$ is a chord of a circle with centre $O$ such that $\angle BAC=30°$. The tangent at $C$ intersects extended $AB$ at a point $D$. Prove that $BC=BD$.

In the adjoining figure, quadrilateral $ABCD$ is circumscribed. Find the value of $x$.

If a hexagon $ABCDEF$ circumscribe a circle, prove that $AB+CD+EF=BC+DE+FA$.

In the adjoining figure, $AB$ is a diameter of a circle. $CP$ is tangent to the circle at $C$. $AD$ and $BE$ are perpendiculars from $A \text{and} B$ respectively on the tangent at $C$. Prove that: $AD + BE= AB$.