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

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

  1. A

    (2+3):23

  2. B

    (2+3):3

  3. C

    (23):23

  4. D

    (23):43

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    Solution:

    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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