# If O is the circumcentre of Triangle ABC and OD prependicular to BC. Prove that Angle BOD = Angle A

O is the circumcentre of ∆ABC and OD⊥BC.

∴ O is the point of intersection of perpendicular bisectors of the sides of ∆ABC.

⇒  D is the mid point of BC.

⇒ BD = DC

In ∆OBD and ∆OCD,

OB = OC  (Radius of the circle)

OD = OD  (Common)

BD = DC  (Proved)

∴ ∆OBD ≅ ∆OCD (SSS Congruence criterion)

⇒∠BOD = ∠COD  (CPCT)

We know that, the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

∴ ∠BOC = 2 ∠BAC

⇒ 2∠BOD = 2 ∠A        [ Using (1) ]

⇒ ∠BOD = ∠A

• 68

plz provide a figure for it

• 5

I too don't hav any figure. You just have to proove it by making your own figure.

• -3

look angle BOD = 1/2 OF angle BOC  (as OD IS THE PERPENDICULAR to BC )...(1)

angle A = 1/2 OF Angle BOC (angle subtended at the centre by the chord is double than the angle subtended by it at any other point on the remaining part of the circle)  ..........(2)

from equation 1 and 2 it is clear that angle BOD = angle A.

HOPE IT HELPS YOU

good luck.......

• 7
What are you looking for?
The passage contains a geometric proof that angle BOD is equal to angle A in a triangle where O is the circumcenter and OD is perpendicular to BC. The proof is provided, and there is a discussion about the need for a figure to accompany the proof.
no_question

the text does not contain a mathematical question but rather a geometric proof and a discussion about the need for a figure to accompany the proof. therefore, the type is no_question.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# If O is the circumcentre of Triangle ABC and OD prependicular to BC. Prove that Angle BOD = Angle A

O is the circumcentre of ∆ABC and OD⊥BC.

∴ O is the point of intersection of perpendicular bisectors of the sides of ∆ABC.

⇒  D is the mid point of BC.

⇒ BD = DC

In ∆OBD and ∆OCD,

OB = OC  (Radius of the circle)

OD = OD  (Common)

BD = DC  (Proved)

∴ ∆OBD ≅ ∆OCD (SSS Congruence criterion)

⇒∠BOD = ∠COD  (CPCT)

We know that, the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

∴ ∠BOC = 2 ∠BAC

⇒ 2∠BOD = 2 ∠A        [ Using (1) ]

⇒ ∠BOD = ∠A

• 68

plz provide a figure for it

• 5

I too don't hav any figure. You just have to proove it by making your own figure.

• -3

look angle BOD = 1/2 OF angle BOC  (as OD IS THE PERPENDICULAR to BC )...(1)

angle A = 1/2 OF Angle BOC (angle subtended at the centre by the chord is double than the angle subtended by it at any other point on the remaining part of the circle)  ..........(2)

from equation 1 and 2 it is clear that angle BOD = angle A.

HOPE IT HELPS YOU

good luck.......

• 7
What are you looking for?
user
The passage contains a geometric proof that angle BOD is equal to angle A in a triangle where O is the circumcenter and OD is perpendicular to BC. The proof is provided, and there is a discussion about the need for a figure to accompany the proof.

TYPE: NO_QUESTION

The text does not contain a mathematical question but rather a geometric proof and a discussion about the need for a figure to accompany the proof. Therefore, the type is NO_QUESTION.