Statement 1: If a,b,c are the sides of a triangle, A is the angle opposite a, and bc = λ 2 with λ 0 , then a ≥ λ sin ( A /2 ) . Statement 2: The arithmetic mean of two positive quantities is at leas

Question

Statement 1: If a,b,c are the sides of a triangle, A is the angle opposite a, and bc = λ 2 with λ > 0 , then a ≥ λ sin ( A /2 ) . Statement 2: The arithmetic mean of two positive quantities is at least their geometric mean.

TestHub · Accepted Answer

The stronger inequality a ≥ 2 λ sin ( A /2 ) follows from the cosine rule: a 2 = ( b − c ) 2 + 4 bc sin 2 ( A /2 ) ≥ 4 λ 2 sin 2 ( A /2 ) . This uses the same AM-GM principle b 2 + c 2 ≥ 2 bc , so Statement 2 explains the weaker claimed bound.

Mathematics - Properties of Triangle Question with Solution | TestHub

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