AB is the diameter and AC is the chord of a circle sucht that angle BAC is 30 degree If then tangent at C intesects AB produced in D prove that BC = BD

Asked by LakshmiSriram | 1st Feb, 2014, 11:10: PM

Expert Answer:

Consider the right triangle, BAC.
 
angle B A C plus angle A C B plus angle A B C equals 180 degree rightwards double arrow 30 degree plus 90 degree plus angle A B C equals 180 degree rightwards double arrow 120 degree plus angle A B C equals 180 degree rightwards double arrow angle A B C equals 180 degree minus 120 degree rightwards double arrow angle A B C equals 60 degree
 
Thus,
 
angle A B C plus angle C B D equals 180 degree rightwards double arrow angle C B D equals 180 degree minus 60 degree rightwards double arrow angle C B D equals 120 degree
 
The angles in the alternate segment are equal.
 
Thus, angle B C D equals 30 degree
By angle sum property,
 
angle C B D plus angle B C D plus angle B D C equals 180 degree rightwards double arrow 120 degree plus 30 degree plus angle B D C equals 180 degree rightwards double arrow 150 degree plus angle B D C equals 180 degree rightwards double arrow angle B D C equals 180 degree minus 150 degree rightwards double arrow angle B D C equals 30 degree S i n c e space angle B C D equals angle B D C equals 30 degree comma space o p p o s i t e space s i d e s space o f space t h e space i s o s c e l e s space t r i a n g l e space a r e space e q u a l. T h u s comma space B C equals B D

Answered by  | 3rd Apr, 2014, 03:33: PM

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