In △ A BC , if I is the incentre then A I + B I + C I ≥ k r then find k

A I = r cosec 2 A , so A I + B I + C I = r ( cosec 2 A + cosec 2 B + cosec 2 C ) . By A.M. ≥ G.M., ∑ cosec 2 A ≥ 3 ( ∏ cosec 2 A ) 1/3 ≥ 3 ( 8 ) 1/3 = 6 , since ∏ sin 2 A ≤ 8 1 . Hence A I + B I + C I ≥ 6 r , so k = 6 .

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Mathematics - Properties of Triangle Question with Solution | TestHub

MathematicsProperties of Triangler1,r–R formula basedMedium2 minQB

MathematicsMediuminteger

In $△ A BC$ , if $I$ is the incentre then $A I + B I + C I \geq k r$ then find $k$

Answer:

Solution:

$A I = r cosec \frac{A}{2}$ , so $A I + B I + C I = r (cosec \frac{A}{2} + cosec \frac{B}{2} + cosec \frac{C}{2})$ . By A.M. $\geq$ G.M., $\sum cosec \frac{A}{2} \geq 3 (\prod cosec \frac{A}{2})^{1/3} \geq 3 (8)^{1/3} = 6$ , since $\prod sin \frac{A}{2} \leq \frac{1}{8}$ . Hence $A I + B I + C I \geq 6 r$ , so $k = 6$ .

Stream:JEESubject:MathematicsTopic:Properties of TriangleSubtopic:r1,r–R formula based

⏱ 2mℹ️ Source: QB

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