AB is a diameter of a circle and AC is its chord such that ∠BAC = 30°. If the tangent at C intersects AB extended at D, then BC = BD. Write ‘True’ or ‘False’ and justify your answer

Solution:

First join OC

∠BCD = ∠BAC = 30° (angles in alternate segment)

Arc BC subtends ∠DOC at the centre of the circle and ∠BAC at the remaining part of the circle.

∠BOC = 2 ∠BAC = 2 × 30° = 60°

In triangle OCD

∠BOC = ∠DOC = 60°

∠OCD = 90°

OC is perpendicular to CD

∠DOC + ∠ODC = 90°

60° + ∠ODC = 90°

∠ODC = 90° - 60° = 30°

In triangle BCD

∠ODC = ∠BDC = ∠BCD = 30°

So BC = BD

Therefore, the statement is true.

✦ Try This: ABCD is a cyclic quadrilateral in which BC is parallel to AD, angle ADC = 110° and angle BAC = 50°. Find angle DAC and angle DCA.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 10

NCERT Exemplar Class 10 Maths Exercise 9.2 Problem 10

AB is a diameter of a circle and AC is its chord such that ∠BAC = 30°. If the tangent at C intersects AB extended at D, then BC = BD. Write ‘True’ or ‘False’ and justify your answer

Summary:

The statement “AB is a diameter of a circle and AC is its chord such that ∠BAC = 30°. If the tangent at C intersects AB extended at D, then BC = BD” is true

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