In a triangle ABC, the length of the bisector of angle A is
- $\dfrac{2bc sin(A/2)}{b + c}$
- $\dfrac{2bc cos(A/2)}{b+c}$
- $\dfrac{abc}{2R(b+c)}cosec\dfrac{A}{2}$
- $\dfrac{4\Delta}{b+c}cosec\dfrac{A}{2}$
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AI explanation
By the angle bisector theorem, the bisector of angle A divides the opposite side BC into segments proportional to the adjacent sides, so we have BE/EC = c/b and BE = ac/(b+c). In triangle ABE, the perpendicular distance from E to AB is given by applying the sine formula as BE sin(B). The length of the angle bisector is the hypotenuse of the right triangle formed by this perpendicular, so its length is BE sin(B) / sin(A/2). Substituting BE and using the sine rule a/sin(A) = b/sin(B), the formula simplifies directly to 2bc sin(A/2) / (b + c).