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# In Two Concentric Circles, Prove that All Chords of the Outer Circle, Which Touch the Inner Circle, Are of Equal Length. - ICSE Class 10 - Mathematics

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ConceptArc and Chord Properties - If Two Chords Intersect Internally Or Externally Then the Product of the Lengths of the Segments Are Equal

#### Question

In two concentric circles, prove that all chords of the outer circle, which touch the inner circle, are of equal length.

#### Solution

i) PQ = RQ
∴ ∠PRQ = ∠QPR (opposite angles of equal sides of a triangle)
⇒ ∠PRQ + ∠QPR  + 68° =  180°
⇒ ∠2 PRQ =  180° -  68°
⇒  ∠PRQ =(112°)/2 = 56°
Now, ∠QOP = 2∠ PRQ (angle at the centre is double)
⇒ QOP = 2 × 56°  = 112°
ii)  ∠PQC = ∠PRQ (angles in alternate segments are equal)
∠QPC = ∠PRQ (angles in alternate segments)
∴  ∠PQC =  ∠QPC = 56° (∵  ∠PRQ =56° from(i))
∠PQC + ∠QPC + ∠QCP = 180°
⇒56° +  56° +  ∠QCP =  180°
⇒ ∠QCP =  68°

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#### APPEARS IN

Solution In Two Concentric Circles, Prove that All Chords of the Outer Circle, Which Touch the Inner Circle, Are of Equal Length. Concept: Arc and Chord Properties - If Two Chords Intersect Internally Or Externally Then the Product of the Lengths of the Segments Are Equal.
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