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In the given figure, AB is a chord of a circle with center O a

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In the given figure, $A B$ is a chord of a circle with center $O$ and $A B$ is produced to $C$ such that $BC = OB$ . Also, $CO$ is joined and produced to meet the circle in $D$ . If $∠ A C D = y^{o}$ and $∠ A O D = x^{o}$ , prove that $x = 3 y$

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In the given figure,

A B

is a chord of a circle with center

O

and

A B

is produced to

C

such that

BC = OB

. Also,

CO

is joined and produced to meet the circle in

D

. If

∠ A C D = y^{o}

and

∠ A O D = x^{o}

, prove that

x = 3 y

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Text solutionVerified

It is given that

A B

is a chord of a circle with centre

O

ans

A B

is produced to

C

such that

BC = OB

We know that

∠ BOC

and

∠ BCO

form isosceles triangle

∠ BOC = ∠ BCO = y^{o}

We know that

O A

and

OB

are the radii of same circle

O A = OB

We know that

∠ O A B

and

∠ OB A

form isosceles triangle

∠ O A B = ∠ OB A = 2 y^{o}

From the figure we know that

Exterior

∠ A O D = ∠ O A C + ∠ A CO

It can be written as

Exterior

∠ A O D = ∠ O A B + ∠ BCO

So we get

Exterior

∠ A O D = 3 y^{o}

It is given that

∠ A O D = x^{o}

So we get

x^{o} = 3 y^{o}

Therefore it is proved that

x = 3 y

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Question Text	In the given figure, $A B$ is a chord of a circle with center $O$ and $A B$ is produced to $C$ such that $BC = OB$ . Also, $CO$ is joined and produced to meet the circle in $D$ . If $∠ A C D = y^{o}$ and $∠ A O D = x^{o}$ , prove that $x = 3 y$
Answer Type	Text solution:1
Upvotes	150

In the given figure, AB is a chord of a circle with center O and AB is produced to C such that BC=OB. Also, CO is joined and produced to meet the circle in D. If ∠ACD=yo and ∠AOD=xo, prove that x=3y

Text solutionVerified

Practice questions from similar books

In the given figure, $A B$ is a chord of a circle with center $O$ and $A B$ is produced to $C$ such that $BC = OB$ . Also, $CO$ is joined and produced to meet the circle in $D$ . If $∠ A C D = y^{o}$ and $∠ A O D = x^{o}$ , prove that $x = 3 y$