Question
If a line intersects two concentric circles (circles with the same centre) with centre $O$ at $A, B, C$ and $D$, prove that $AB = CD.$
✓
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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