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Question

If r is the radius and R is the circumradius. Then 2Rr

A
True
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B
False
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Solution

The correct option is B False
r=4RsinA2sinB2sinC2Now consider,cos(A)+cos(B)+cos(C)=2cos((A+B)2)cos((AB)2)(2cos2((A+B)2)1)=2(cos((AB)2))cos((A+B)2)cos((A+B)2)+1=4sin(A2)sin(B2)sin(C2)+1
It is useful to remember the identity, cos(A) + cos(B) + cos(C) 32 though the proof is a little beyond the scope of this chapter
Therefore, 4sin(A2)sin(B2)sin(C2)+132SinA2SinB2SinC212.12.12=18r<4R(18)=R2
Given statement is false

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