# Ex.10.4 Q4 Circles Solution - NCERT Maths Class 9

## Question

If a line intersects two concentric circles (circles with the same center) with centre \({O}\) at \(A, B, C\) and \(D\), prove that \({AB = CD.}\)

## Text Solution

**What is known?**

Two concentric circles with centre \(\begin{align} {O.} \end{align}\)

**What is unknown?**

Proof that \({AB = CD}\)

**Reasoning:**

Perpendicular drawn from the centre of the circle bisects the chord.

**Steps:**

Draw a perpendicular from the centre of the circle \({OM}\) to the line \({AD}\).

We can see that \({BC}\) is the chord of the smaller circle and \({AD}\) is the chord of the bigger circle.

We know that perpendicular drawn from the centre of the circle bisects the chord.

\(\begin{align}∴ {BM}={MC} \ldots(1)\end{align}\)

And, \(AM = MD ... (2)\)

Subtracting (**\(2\)**) from (**\(1\)**), we obtain

\[\begin{align}{AM - BM }= {MD} - {MC}\end{align}\]

\[\begin{align}∴ {AB = CD}\end{align}\]