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State True or False.
In the given figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Then $Δ P A C ∼ Δ P B D$ .

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True

False

answer is B.

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Detailed Solution

Given figure is,
Question Image

ABCD is a cyclic quadrilateral.
In cyclic quadrilateral, opposite angles are supplementary. So,

∠ A C D + ∠ A B D = 180 ∘

[1]
Also PD is a line, so all angles on that line form linear pair. So,

∠ P C A + ∠ A C D = 180 ∘

[2]
From [1] and [2],

∠ P C A = ∠ A B D

[3]
In

Δ P A C and Δ P B D

∠ A P C = ∠ D P B

[common]

∠ P C A = ∠ P B D

[from 3]
By AA similarity,

Δ P A C ∼ Δ P D B .

Our assertion is not the same as our solution i.e., the given statement is false.
In the given figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Then

Δ P A C ∼ Δ P D B .

Hence the correct option is 2.

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