# Ex.10.6 Q1 Circles Solution - NCERT Maths Class 9

## Question

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

## Text Solution

**What is known?**

Two intersecting circles.

**What is unknown?**

To prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection

**Reasoning:**

Using Side-Side-Side (**SSS criteria**) and Corresponding parts of congruent triangles (**CPCT**) we prove the statement.

**Steps:**

Draw **\(2\)** intersecting circles with centres \({O}\) and \({O’}\) respectively. Join these **\(2\)** centres.

Let the points of intersection be \({A}\) and \({B.}\)

We need to prove that

\(\begin {align}\angle {OAO}^{\prime}=\angle {OBO}^{\prime} \end {align}\)

Consider \(\begin {align}\Delta {OAO}^{\prime} \end {align}\) and \(\begin {align}\Delta {OBO}^{\prime} \end {align}\)

\[\begin{align}{OA}&={OB}\\ \text { (Radii of circle}&\text{ with centre $O$)}\\\\ {O}^{\prime} {A}&={O}^{\prime } {B} \\ \text { (Radii of circle} & \text {with centre $O^\prime $)} \\\\{OO}^{\prime}&={O} {O}^{\prime} \\&\text { (Common) }\end{align}\]

Therefore by **SSS** criteria, \(\begin {align}\Delta {O} {AO}^{\prime} \end {align}\) and \(\begin {align}\Delta {O} {BO}^{\prime} \end {align}\) are congruent to each other.

\[\begin {align}{\rm{By\,\, CPCT}} , \;\angle {OAO}^{\prime}=\angle {OBO}^{\prime} \end {align}\]

Hence it is proved that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.