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jai kohli
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Asked: 2023-02-16T03:52:49+00:00 In:

Two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that (i) ∆ PAC ~ ∆ PDB (ii) PA . PB = PC . PD

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How to find the vertical angle.
Prove the congruent by the hypotenuse-leg congruence theorem.
Class 10th Maths, Page No:153, Important Triangles Solutions, Exercise 6.6, Questions No:8.
CBSE Board and Others state Board. Session 2022-2023.

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      Two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that (i) ∆ PAC ~ ∆ PDB (ii) PA . PB = PC . PD

      Join CB.
      (i) In ΔAPC and DPB,
      ∠APC = ∠DPB [Vertically Opposite Angles]
      ∠CAP = ∠BDP [Angles in the some segment]
      ΔAPC ∼ Δ DPB [ AA similarity]

      (ii) We have already proved that ΔAPC ∼ ΔDPB.
      We know that the corresponding sides of similar triangle are proportional.
      So, AP/DP = PC/PB = CA/BD ⇒ AP/DP = PC/PC ⇒ AP. PB = PC . DP

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