Triangles And Properties Of Triangle Question 30
Question: O is the circumventer of the triangle ABC and $ R_1,R_2,R_3 $ are the radii of the circumcircles of the triangles OBA, OCA and OAB respectively, then $ \frac{a}{R_1}+\frac{b}{R_2}+\frac{c}{R_3} $ is equal to
Options:
A) $ \frac{abc}{R} $
B) $ \frac{abc}{R^{3}} $
C) $ \frac{abc}{R^{4}} $
D) None
Show Answer
Answer:
Correct Answer: B
Solution:
[b] $ R_1=\frac{BC}{2\sin (\angle BOC)}=\frac{a}{2\sin 2A} $
$ \therefore \frac{a}{R_1}=2\sin 2A $ Similarly, $ \frac{b}{R_2}=2\sin 2B $ and $ \frac{c}{R_3}=2\sin 2C $ So, $ \frac{a}{R_1}+\frac{b}{R_2}+\frac{c}{R_3} $ $ =2(sin2A+sin2B+sin2C) $ $ =2.4\sin A\sin B\sin C $ $ [\because ,A+B+C=\pi ] $ $ =(2sinA)(2sinB)(2sinC)=( \frac{a}{R} )( \frac{b}{R} )( \frac{c}{R} ) $
BETA
