In any Δ ABC, we known the relationship between the sides (a, b, c), angles (A,B,C), circum-radius (R), area ( Δ ) and other parameters.
= 
= 
= 
= 
Using the above information, answer the following questions-
(i) If two sides of a triangle are 5 and 8, and its circum radius is 
, then the third side can be-
Text Solution
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Ans.
(i)
Sol. R = 
⇒ sin A = 
= 
,
a = 5, b = 8, R = 25/6
∴ cos A = 4/5 or – 4/5
∴ cos A = 
∴ 
= 
⇒ c 2 – 64c + 195 = 0
⇒ c = 5, 7.8.
(ii)
Sol. a cos A + b cos B + c cos C
= 2R sin A cos A + 2R sin B cos B + 2R sin C cos C
= R {sin 2A + sin 2B + sin 2C}
= 4R sin A sin B sin C
(iii)
Sol. 8R 2 = a 2 + b 2 + c 2 ⇒ 8R
2 = 4R 2 sin 2 A + 4R 2 sin 2 B + 4 sin 2 C
⇒ 2 = 
+ 
+ 
⇒ cos 2A + cos 2B + cos 2C + 1 = 0
⇒ –1 – 4 cos A cos B cos C + 1 = 0
⇒ cos A. cos B. cos C = 0
⇒ A = 
or B = 
or C = 
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