Question
If a line intersects two concentric circles $($circles with the same centre$)$ with centre $O$ at $A , B , C$ and $D ,$ prove that $A B=C D$.
✓
Answer
Given: $A$ line intersects two concentric circles $($circles with the same centre$)$ with centre $O$ at $A, B, C$ and $D.$
To prove : $AB = CD$
Construction : Draw $O M \perp B C$
Proof: $\because$ The perpendicular drawn from the centre of a circle to a chord bisects the chord.
$\therefore AM = DM$
and $ BM = CM $
Subtracting $(2)$ from $(1),$ we get
$ AM - BM = DM - CM$
$\Rightarrow AB = CD $
To prove : $AB = CD$
Construction : Draw $O M \perp B C$
Proof: $\because$ The perpendicular drawn from the centre of a circle to a chord bisects the chord.
$\therefore AM = DM$
and $ BM = CM $
Subtracting $(2)$ from $(1),$ we get
$ AM - BM = DM - CM$
$\Rightarrow AB = CD $
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