Challenge your brain 1
Geometry
Level
5
If in a triangle ABC , a^4+b^4+c^4 = 2(ab)^2+2(ca)^2+(bc)^2, then angle BAC = ?
The answer is 30.
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1 solution
In triangle ABC, a^4+b^4+c^4 = 2(ab)^2+2(ca)^2+(bc)^2. Consider a^4+b^4+c^4 - 2(ab)^2+2(ca)^2 = (bc)^2. Factorising a^4+b^4+c^4 - 2(ab)^2+2(ca)^2 = (bc)^2 as (a^2 - b^2 - c^2)^2 - 2(bc)^2 = (bc)^2. (a^2 - b^2 - c^2)^2 = 3(bc)^2. (a^2 - b^2 - c^2) = +sqrt(3)(bc) or -sqrt(3)(bc). Rearranging: b^2 + c^2 - a^2 = +sqrt(3)(bc) or -sqrt(3)(bc). (b^2 + c^2 - a^2)/(bc) = +sqrt(3) or -sqrt(3). (b^2 + c^2 - a^2)/2(bc) = +sqrt(3)/2 or -sqrt(3)/2. (b^2 + c^2 - a^2)/2(bc) = cos A = 30 degrees or 150 degrees.