Maths · Determinants and matrices of order two and three
In a triangle with usual notations, if then is equal to
In a triangle \( A B C, \) with usual notations, if \( \left|\begin{array}{ccc}1 & a & b \\ 1 & c & a \\ 1 & b & c\end{array}\right|=0, \) then \( 4 \sin ^{2} A+ \) \( 24 \sin ^{2} B+36 \sin ^{2} C \) is equal to
- A. 48
- B. 50
- C. 44
- D. 34
Step-by-step solution
The given determinant equated to zero simplifies to a^2 + b^2 + c^2 = ab + bc + ca, which implies (a-b)^2+(b-c)^2+(c-a)^2=0, so a=b=c. Thus triangle ABC is equilateral with angles A=B=C=60°. Then sin60°=√3/2, so 4 sin^2 A + 24 sin^2 B + 36 sin^2 C = 4*(3/4) + 24*(3/4) + 36*(3/4) = 48.
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