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If the angles of a right angled triangle are in AP., then the ratio of the in-radius and the perimeter, is

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a
(2+3):23
b
(2+3):3
c
(23):23
d
(23):43

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detailed solution

Correct option is C

Let ABC be a right angled triangle right angle at B.

Let the other angles be A=90d and C=902d. Then

A+B+C=1802703d=180d=30A=60 and C=30

Let AC =b. Then,

c=AB=ACcos60=b2 and a=BC=bcos30=32 2s=a+b+c=3b2+b+b2=(3+3)b2

Also,

Δ=12(BC×AB)=123b2×b2=3b28

 Require ratio =r2s=Δ2s2

=3b282(3+3)24b2=3(3+3)2=33(3+1)2=(31)243=42343=2323

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