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23. Two circles having their centres at O and P intersect each
other at Q and R. Two straight lines AQB and CRD are
drawn parallel to OP. Prove that
(a) AB = 20P, and
(b) AB =CD.
Answers
Solution :-
Provided
• AB || OP || CD
(a) To prove 2 OP = AB
Const:-
• OM and PN perpendicular to AB.
• Join QR and Mark point S where it intersect OP.
Proof :-
AQ is a chord , hence the perpendicular from center will bisect it.
→ 2 OS = AQ ...(i)
QB is a chord , hence the perpendicular from center will bisect it.
→2 SP = QB ....(ii)
Add (i) & (ii)
→ 2 ( OS + SP ) = AQ + QB
→ 2 OP = AB
(b) To prove AB = CD
Const:-
• Join QR and Mark point S where it intersect OP.
• Join OQ , OR , PR & PQ
Here
• QS = height of ∆OPQ
• RS = height of ∆OPR
Now in ∆OPQ & ∆OPR
• OP = OP ( common)
• OQ = OR ( raddii)
• PQ = PR (raddii)
Hence ∆OPQ ≅ ∆OPR
Therefore , SQ = SR ( cpct)
Now distance of
• Chord AQ from Center = Chord CR from center
→ AQ = CR ( property of circle that Chords equidistant from center have same length) ...(i)
• Chord QB from Center = Chord RD from center
→ QB = RD ( property of circle that Chords equidistant from center have same length) ...(ii)
Add (i) & (ii)
→ AQ + QB = CR + RD
→ AB = CD
Step-by-step explanation:
Hope it helps!!!!!!!!!