Question

$\mid \overline \mathbf \purple{hy \: guys}$


Let the vertex of an angle ABC be located outside a circle and let the sides intersect equal chords AD and CE with the circe. Prove that angle ABC is equal to half of the difference of the angle sublended by the chords AC and DE at the centre.​

Answer 1

[Refer the attachment for figure]

$\implies\:\sf{\angle{AEC}\:=\:x}$

At centre O, it will be twice of angle AEC.

$\implies\:\sf{\angle{AOC}\:=\:2x}$

Angle ABC is equal to half of the difference of the angle sublended by the chords AC and DE.

$\implies\:\sf{\angle{DAE}\:=\:y}$

At centre it will be twice of angle DAE.

$\implies\:\sf{\angle{DOE}\:=\:2y}$

Now, BEA is a line means of 180°.

In ∆ABE

By angle sum property,

$\implies\:\sf{\angle{B}+y+180^{\circ}-x\:=\:180^{\circ}}$

180° throughout cancel,

$\implies\:\sf{\angle{B}+y-x\:=\:0}$

$\implies\:\sf{\angle{B}\:=\:x-y}$

Now, we can write x as angle ACE and y as angle DAE.

$\implies\:\sf{\angle{B}\:=\:\angle{ACE}-\angle{DAE}}$

Angle ACE = 1/2 angle AOC and angle DAE = 1/2 angle DOE

$\implies\:\sf{\angle{B}\:=\:\frac{1}{2}\angle{AOC}-\frac{1}{2}\angle{DOE}}$

$\implies\:\sf{\angle{B}\:=\:\frac{1}{2}(\angle{AOC}-\angle{DOE})}$

Here, angle AOC is difference of angle subtended by cord AC at centre O and D of angle DOE at O.

Hence, proved

Answer 2

$\underline \mathbb{SOLUTION}$

》The angles Subtended by an Arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

$\bold\green{\underline{We\:know}}$

》Exterior angle of a triangle is equal to the sum of the interior opposite angles.

$\boxed{IN\:ΔBDC}$

》∠ADC = ∠DBC + ∠DCB ––––– (1)

$\bold\green{\underline{We\:know}}$

About the angle at the centre is twice the angle at that point on the remaining part of the circle.

》∠DCE = 1/2 ∠DOE

》∠DCB = 1/2 ∠DOE

》[∠DCE = ∠DCB]

》∠ADC = 1/2 ∠AOC

Putting values of ∠ADC and ∠DCB in eqn (1)

》1/2 ∠AOC = ∠ABC + 1/2 ∠DOE

》[∠DBC = ∠ABC]

》∠ABC = 1/2 (∠AOC - ∠DOE)

$\bold\green{\underline{Hence,}}$

∠ABC is equal to Have the difference of the angle subtended by chords AC and DE at the centre.

NOTE:

Refer to attachment for the figure

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$\bold\blue{\underline{Hence\:Proved}}$