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Standard IX

Mathematics

Perpendicular from the Center to a Chord Bisects the Chord

If a line int...

Question

If a line intersects two concentric circles with centre $A$ in points $P, Q, R$ and $S$ respectively, then prove that $P Q = R S$ .

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Solution

To prove:
$P Q = R S$
Proof:
We know that when a perpendicular is drawn from the centre of the circle on a chord, it bisects the chord.
$Q T = R T$ and $. . . . . . . . . . . . . . . . . (i)$
$P T = S T$ $. . . . . . . . . . . . . . . . (i i)$
Now,
Subtract equation $(i)$ from equation $(i i)$ :
$P T - Q T = S T - R T$
or, $P Q = R S$
Hence, Proved.

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If a line intersects two concentric circles with centre A in points P,Q,R and S respectively, then prove that PQ=RS.

To prove:PQ=RSProof:We know that when a perpendicular is drawn from the centre of the circle on a chord, it bisects the chord.QT=RT and .................(i)PT=ST ................(ii)Now, Subtract equation (i) from equation (ii):PT−QT=ST−RTor, PQ=RSHence, Proved.

If a line intersects two concentric circles with centre $A$ in points $P, Q, R$ and $S$ respectively, then prove that $P Q = R S$ .