MHT CET · Maths · Properties of Triangles
In a triangle ABC , with usual notations, \(2 \mathrm{ac} \sin \left(\frac{\mathrm{A}-\mathrm{B}+\mathrm{C}}{2}\right)\) is equal to
- A \(a^2+b^2-c^2\)
- B \(b^2-a^2+c^2\)
- C \(\mathrm{c}^2+\mathrm{a}^2-\mathrm{b}^2\)
- D \(\mathrm{a}^2-\mathrm{b}^2-\mathrm{c}^2\)
Answer & Solution
Correct Answer
(C) \(\mathrm{c}^2+\mathrm{a}^2-\mathrm{b}^2\)
Step-by-step Solution
Detailed explanation
\(2 \mathrm{ac} \sin \frac{\mathrm{A}-\mathrm{B}+\mathrm{C}}{2} =2 \mathrm{ac} \sin \frac{\pi-2 \mathrm{~B}}{2} \)
\( =2 \mathrm{ac} \cos \mathrm{B} \)
\( =2 \mathrm{ac} \frac{\mathrm{c}^2+\mathrm{a}^2-\mathrm{b}^2}{2 \mathrm{ca}} \)
\( =\mathrm{c}^2+\mathrm{a}^2-\mathrm{b}^2\)....[By cosine rule]
\( =2 \mathrm{ac} \cos \mathrm{B} \)
\( =2 \mathrm{ac} \frac{\mathrm{c}^2+\mathrm{a}^2-\mathrm{b}^2}{2 \mathrm{ca}} \)
\( =\mathrm{c}^2+\mathrm{a}^2-\mathrm{b}^2\)....[By cosine rule]
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