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In !ABC it is given that a:b:c = cos A:c...

In `!ABC` it is given that a:b:c = cos A:cos B:cos C
Statement-1: `!ABC` is equilateral.
Statement-2: cosA`=(b^(2)+c^(2)-a^(2))/(2bc),cosB=(c^(2)+a^(2)-b^(2))/(2ac),cosC=(a^(2)+b^(2)-c^(2))/(2ab)`

A

Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.

B

Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.

C

Statement-1 is True, Statement-2 is False.

D

Statement-1 is False, Statement- 2 is True.

Text Solution

Verified by Experts

Clearly, statement-2 is true.
Now, a:b: c =cos A: cos B:cosC
`rArra/(cosA)=b/(cosB)=c/(cosC)`
`rArr(2abc)/(b^(2)+c^(2)-a^(2))=(2abc)/(c^(2)+a^(2)-b^(2))=(2abc)/(a^(2)+b^(2)-c^(2))`
`rArrb^(2)+c^(2)-a^(2)=c^(2)+a^(2)-b^(2)=a^(2)+b^(2)-c^(2)`
`rArrb^(2)-a^(2)=a^(2)-b^(2)` and `c^(2)-b^(2)=b^(2)-c^(2)`
`rArra^(2)=b^(2)=c^(2)rArra=b=crArr!ABC` is equilateral.
So statement-1 is also true and statement-2 is a correct explanation for statement-1.
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