Multiple choice

Write True or False and justify your answer in each of the following : AB is a diameter of a circle and AC is its chord such that $\angle BAC =30^{0}$. If the tangent at C intersects AB extended at D, then BC $=$ BD.

  1. True

  2. False

  3. Ambiguous

  4. Data Insufficient
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In triangle ABC, angle BAC = 30. Since AB is diameter, angle ACB = 90. Thus angle ABC = 60. Tangent at C makes angle with chord AC equal to angle in alternate segment, so angle BCD = 30. In triangle BCD, angle CBD = 180 - 60 = 120. Angle BCD = 30, so angle BDC = 180 - 120 - 30 = 30. Since two angles are 30, BC = BD.

AI explanation

Since AB is the diameter, angle ACB is 90 degrees by Thales's theorem. In triangle ABC, angle BAC is given as 30 degrees, which means angle ABC must be 60 degrees, making angle CBD 120 degrees. Since CD is a tangent and OC is a radius, angle OCD is 90 degrees, meaning angle COD is 60 degrees; the two radii OC and OB make triangle OCB isosceles, so angle OCB is also 60 degrees. This leaves angle BCD as 30 degrees, making triangle BCD isosceles with angle CBD equal to 120 degrees and the other two angles 30 degrees, so the sides opposite those equal angles are equal (BC = BD).

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