Question

# Two sides of a triangle are 2 and √3 and the included angle is 30′ then the in-radius r of the triangle is equal

A

14(31)

B

12(3+1)

C

12(31)

D

14(3+1)

Solution

## The correct option is C 12(√3−1) We know that r=(s−a)tanA2. Let the given sides be b and c and A=30′ then  a2=b2+c2−2bccosA by cosine rule =4+3−4√3cos30′a2=7−4√3×√22=7−6=1∴a2=1⇒a=12s=a+b+c=1+2+√3=3+√3∴2s−2a=3+√3−2⇒(s−a)=√3+12A=30′⇒A2=15′∴tanA2=tan15′=√3−1√3+1∴r=(s−a)tanA2=√3+12.√3−1√3+1=12(√3−1)

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