Question

Ab is a diameter of a circle with centre o and radius od is perpendicular to ab . If c is any point on arc db, find angle bad and angle acd

Answer 1

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Vinayak Gupta asked in Math

AB is a diameter of a cicle with centre o and radius od is perpendicular to AB. if c is any point on arc DB, find angle BAD and angle ACD.

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Gopal.mohanty... answered this
26023 helpful votes in Math, Class XII-Science

Dear Student!

Here are the answers to your question.

 

 

 

In ∆AOD and ∆BOD

OA = OB  (radii of same circle)

OD = OD  (common)

∠AOD = ∠BOD  (each 90°)

∴ ∠AOD ≅ ∆BOD  (SAS congruency)

⇒ ∠BAD = ∠ABD  (C.P.C.T)  … (1)

 

Angle in a semicircle is a right angle

∴ ∠ADB = 90°

Using angle sum properly in ∆ABD; ∠ABD + ∠BAD + ∠ADB = 180°

⇒∠BAD + ∠BAD + 90° = 180°

⇒2∠BAD = 180° – 90° = 90°

⇒ ∠BAD = 45°

∴ ∠ABD = 45°  [From (1)]

∠ABD and ∠ACD are the angles in the segment AD

∴ ∠ACD = ∠ABD

⇒ ∠ACD = 45°

Answer 2

∠BAD = 45° and ∠ACD = 45°

GIVEN

AB is a diameter of a circle with centre O and radius OD is perpendicular to AB. C is any point on arc DB.

TO FIND

∠BAD and ∠ACD

SOLUTION

We can simply solve the above problem as follows;

In ΔAOD and BOD

OA = OB (Radii of the same circle)

OD = OD (Common side)

∠AOD = ∠BOD (90°)

By SAS Congruency,

ΔAOD ≈ ΔBOD

By CPCT;

∠BAD = ∠ABD

We know that,

Angle in a semicircle is 90°.

Therefore,

∠ABD = 90°

In ΔABD

∠ABD + ∠BAD + ∠ADB = 180

2∠BAD + 90 = 180

2∠BAD = 180 - 90

2∠BAD = 90

∠BAD = 45°

∠ABD = 45°

∠ACD = ∠ABD = 45°

Hence, ∠BAD = 45° and ∠ACD = 45°

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