If A- B- C- D are four points such that - BAC- 30 - - BDC- 60 - then D is the center of the circle through A- B and C- The statement is

The correct option is A sometimes true #160;Given A, B, C & D are four points such that ∠ BAC= 30 ^ o & ∠ BDC= 60 ^ o . ...

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Standard VIII

Mathematics

Introduction to Curves

If A, B, C, D...

Question

If A, B, C, D are four points such that $∠ B A C = 30^{\circ}$ & $∠ B D C = 60^{\circ},$ then D is the center of the circle through A, B and C. The statement is

sometimes true

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false

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true

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none of the above.

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Solution

The correct option is A sometimes true
$G i v e n - A, B, C & D a r e f o u r p o i n t s s u c h t h a t ∠ B A C = 30^{o} & ∠ B D C = 60^{o} . T o f i n d o u t - i f D i s t h e c e n t r e o f t h e c i r c l e w h i c h p a s s e s t h r o u g h A, B & C . S o l u t i o n - ∠ B A C = 30^{o} & ∠ B D C = 60^{o} . ∴ ∠ B D C = 2 ∠ B A C . W e k n o w t h a t t h e a n g l e s u b t e n d e d b y a c h o r d a t t h e c e n t r e o f a c i r c l e i s d o u b l e t h e a n g l e a t t h e c i r c u m f e r e n c e s u b t e n d e d b y t h e s a m e c h o r d . ∴ ∠ B D C i s t h e a n g l e a t t h e c e n t r e a n d ∠ B A C i s t h e a n g l e a t t h e c i r c u m f e r e n c e s u b t e n d e d b y t h e c h o r d B C . S o D i s t h e c e n t r e o f t h e c i r c l e w h i c h p a s s e s t h r o u g h A, B & C . ∴ T h e s t a t e m e n t i s c o r r e c t b y f i g I B u t b y f i g I I t h i s i s n o t t r u e . S o t h e s t a t e m e n t i s n o t a l w a y s t r u e . A n s - O p t i o n A .$

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Similar questions

Q. If A, B, C, D are four points such that

∠ B A C = 30^{\circ} a n d ∠ B D C = 60^{\circ}

then

prove that

D is the centre of the circle through A, B and C

Q. Question 8
Write True or False and justify your answer :

If A, B, C and D are four points such that

∠ B A C = 30^{\circ} a n d ∠ B D C = 60^{\circ}

then D is the centre of the circle through A, B, C and D.

Q. If A, B, C and D are four points such that

∠ B A C = 45^{\circ} a n d ∠ B D C = 45^{\circ}

then A, B, C, D are concyclic

Q. Question 9
Write True or False and justify your answer :

If A, B, C and D are four points such that

∠ B A C = 45^{\circ}

and

∠ B D C = 45^{\circ}

, then A, B, C, D are concyclic.

In the given figure, A, B and C are three points on the circle with centre O such that $∠ B O C = 30^{\circ}$ and $∠ A O B = 60^{\circ}$ . If D is a point on the circle that is not on the arc ABC, then find $∠ A D C .$

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If A, B, C, D are four points such that ∠BAC=30∘ & ∠BDC=60∘, then D is the center of the circle through A, B and C. The statement is

If A, B, C, D are four points such that $∠ B A C = 30^{\circ}$ & $∠ B D C = 60^{\circ},$ then D is the center of the circle through A, B and C. The statement is