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Standard VII

Mathematics

Perpendicular Bisector and It's Construction

O is the circ...

Question

$O$ is the circumcentre of the triangle $A B C$ and $O D$ is perpendicular on $B C$ . Prove that $∠ B O D = ∠ A$ .

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Solution

in triangle BOC and triangle COD

BO = CO (both are radius)

∠BDO=∠CDO =90°

BD is common in both triangle

so both triangle are similar

so ∠BOD=∠COD

as we know angle at the center is double of the angle of corcumferece

∠BOC=2∠BAC

⇒2∠BOD=2∠BAC (as we proved above)

⇒∠BOD=∠A

hence proved.

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Similar questions

In the figure, O is the circumcentre of the triangle ABC, AB = AC and the line OD is perpendicular to the side BC. If BC = 24 cm and OD = 5 cm, then find the circumradius.

Q. Question 7
O is the circumcentre of the

Δ A B C

and D is the mid-point of the base BC. Prove that

∠ B O D = ∠ A

O

is the centroid of

△ A B C

and

D

is mid point of base

B C

then prove

∠ B O D = ∠ A .

Q. Fill In The Blanks

If O is the circumcentre of ΔABC and D is the mid-point of the base BC, then ∠BOD = _______________.

Q. In an acute-angled triangle

A B C,

a point

D

lies on the segment

B C .

Let

O_{1}, O_{2}

denote the circumcentres of

Δ A B D

and

Δ A C D,

respectively. Prove that the line joining the circumcentre of

Δ A B C

and the orthocentre of

Δ O_{1} O_{2} D

is parallel to

B C .

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O is the circumcentre of the triangle ABC and OD is perpendicular on BC. Prove that ∠BOD=∠A.

in triangle BOC and triangle CODBO = CO (both are radius)∠BDO=∠CDO =90°BD is common in both triangleso both triangle are similarso ∠BOD=∠CODas we know angle at the center is double of the angle of corcumferece∠BOC=2∠BAC⇒2∠BOD=2∠BAC (as we proved above)⇒∠BOD=∠Ahence proved.

$O$ is the circumcentre of the triangle $A B C$ and $O D$ is perpendicular on $B C$ . Prove that $∠ B O D = ∠ A$ .