 Hello students, let's work out the following problem. It says AB is a diameter and AC is a chord of a circle such that angle BAC is 30 degrees. If the tangent at C intersects AB produced in D, prove that BC is equal to VD. So we have given a circle in which AB is the diameter, AC is the chord such that angle BAC is 30 degrees and tangent at C produced in D intersects AB. We have to prove that BC is equal to BD. So let's now move on to the solution. What we have to prove? We have to prove that BC is equal to BD and we are given that angle BAC is 30 degrees. Now angle CB is 90 degrees because it's an angle in semicircle. Now let this be angle 1, this be angle 2, this be angle 3, this be 4. 4 is equal to angle BAC plus angle 1 since it's the exterior angle of triangle ABC at and also angle BAC is 30 degrees and angle 1 is 90 degrees so angle 4 is 120 degrees and angle BAC is equal to angle 2 since these are the angles in alternate segments that is angle 2 is 30 degrees since angle BAC is 30 degrees. Now we know that the sum of three angles in a triangle is 180 degrees so now we have angle 2 plus angle 3 plus angle 4 is equal to 180 degrees angle some property of triangle now angle 2 is 30 degrees plus angle 3 plus angle 4 which is 120 degrees so this implies angle 3 is equal to 180 degrees minus 120 degrees minus 30 degrees that is 30 degrees so we have got that angle 2 is equal to angle 3 is equal to 30 degrees and by the property of isosceles triangle sides opposite the equal angles are equal so this implies BC is equal to VD that is sides opposite the equal angles equal now the side opposite the angle 3 is BC and side opposite to the angle 2 is VD so BC is equal to VD because we have proved the required so this completes the question and the session why for now take care have a good day