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Question

In a non-right-angled triangle  PQR, let p,q,r denote the lengths of the sides opposite to the angles at P,Q,R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and PE intersect at O. If p=3,q=1, and the radius of the circumcircle of the  PQR equals 1, which of the following options is/are correct?
  1. Length of RS = 72
  2. Area of  SOE=312
  3. Length of OE = 16
  4. Radius of incircle of  PQR=32(23)


Solution

The correct options are
A Length of RS = 72
C Length of OE = 16
D Radius of incircle of  PQR=32(23)

Using Sine law,
PRsinQ=QRsinP=2R=2
PR=q=1,  p=QR=3
sinP=32P=π3 or 2π3
sinQ=12Q=π6 or 5π6
Since p>qP>Q
Hence, P=2π3 and Q=R=π/6
PQ=2RsinR=2.1.sinπ6=1
Using Apollonius' theorem
2((RS)2+(PS)2)=(PR)2+(RQ)2
2((RS)2+(12)2)=(1)2+(3)2
Therefore, Length of RS=72

Radius of incircle r=4RsinP2.sinQ2.sinR2=4.1.32sin215o
r=32(23)

From Fig. we can observe that,
(PE)2+(RE)2=(PR)2
(PE)2=134
PE=12
From PQR, using property 
OEPE=13OE=16

Now,
area of SEF=14× area of PQR
=14.12.3.12=316

area of SOE=13× area of SEF=112× area of PQR
=112.12.3.12=348

 

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