# Ex.10.4 Q1 Circles Solution - NCERT Maths Class 9

## Question

Two circles of radii \(5 \rm{cm}\) and \(3 \rm{cm}\) intersect at two points and the distance between their centers is \(4 \rm{cm}.\) Find the length of the common chord.

## Text Solution

**What is known?**

Radii of two circles and distance between the centers of the circles.

**What is unknown?**

Length of common chord.

**Reasoning:**

Perpendicular bisector of the common chord passes through the centers of both the circles.

**Steps:**

Given that the circles intersect at \(2\) points, so we can draw the above figure.

Let \(AB\) be the common chord. Let \(O\) and \(O^\prime\) be the centres of the circles respectively.

\(\begin{align} &{O}^{\prime} {A}=5 {\rm{cm}}, {OA}=3 {\rm{cm}},\\ & {OO}^{\prime}=4 {\rm{cm}}\end{align}\)

Since the radius of the bigger circle is more than the distance between the \(2\) centres, we can say that the centre of smaller circle lies inside the bigger circle itself.

\({OO}’\) is the perpendicular bisector of \({AB.}\)

\(\begin{align} {\rm{So,}} \;{OA}&={OB}=3 \,\rm{cm}\\ {AB}&=3+3=6\, \rm{cm} \end{align}\)

Length of the common chord is \(6 \,\rm{cm.}\)

It is also evident that common chord is the diameter of the smaller circle.