Question

Given. In a circle with centre B arc APC =
arc DQE
To prove : Chord AC≈ chord DE​

Answer 1

$\large\bf{\underline\red{Good \: Afternoon♡}}$

Question :-

Theorem : The chords corresponding to congruent arcs of a circle (or congruent circles) are congruent.

Answer :-

Given: B is the centre of circle.

arc APC ≅ arc DQE

To prove: chord AC ≅ chord DE

Proof: [m(arc APC) = ∠ABC (i) [Definition of measure of m(arc DQE) = ∠DBE] (ii) minor arc]

arc APC ≅ arc ∠DQE (iii) [Given]

∴ ∠ABC ≅ ∠DBE [From (i), (ii) and (iii)]

In ∆ABC and ∆DBE,

side AB ≅ side DB [Radil of the same circle]

side [CB] side [EB] [Radii of the same circle]

∠ABC ≅∠DBE [From (iii), Measures of congruent arcs]

∴ ∆ABC ≅ ∆DBE [SAS test of congruency]

∴ chord AC ≅ chord DE [c.s.c.t]

Answer 2

Step-by-step explanation:

Given: B is the centre of circle.

arc APC ≅ arc DQE

To prove: chord AC ≅ chord DE

Proof: [m(arc APC) = ∠ABC (i) [Definition of measure of m(arc DQE) = ∠DBE] (ii) minor arc]

arc APC ≅ arc ∠DQE (iii) [Given]

∴ ∠ABC ≅ ∠DBE [From (i), (ii) and (iii)]

In ∆ABC and ∆DBE,

side AB ≅ side DB [Radil of the same circle]

side [CB] side [EB] [Radii of the same circle]

∠ABC ≅∠DBE [From (iii), Measures of congruent arcs]

∴ ∆ABC ≅ ∆DBE [SAS test of congruency]

∴ chord AC ≅ chord DE [c.s.c.t]