Question

dante is proving the law of cosines for an obtuse triangle. what is the next step of the proof? prove: ( a^{2}=b^{2}+c^{2}-2bccos a ), for an obtuse triangle. 1. draw an altitude from vertex b. the altitude forms two right triangles, ( \triangle bcd ) and ( \triangle abd ). 2. by the pythagorean theorem, ( c^{2}=h^{2}+x^{2} ) and ( a^{2}=h^{2}+(b + x)^{2} ). 3. expanding the binomial squared, ( a^{2}=h^{2}+b^{2}+2bx+x^{2} ). subtract the ( c^{2} ) equation from the ( a^{2} ) equation, resulting in ( a^{2}-c^{2}=b^{2}+2bx ). add together the two equations, resulting in ( a^{2}+c^{2}=2h^{2}+b^{2}+2x^{2}+bx ). add together the two equations, resulting in ( a^{2}+c^{2}=2h^{2}+b^{2}+2x^{2}+2bx ). subtract the ( c^{2} ) equation from the ( a^{2} ) equation, resulting in ( a^{2}-c^{2}=b^{2}+bx ).

Explanation:

Step1: Subtract the \(c^{2}\) equation from the \(a^{2}\) equation

We have \(a^{2}=h^{2}+b^{2}+2bx + x^{2}\) and \(c^{2}=h^{2}+x^{2}\).
Subtracting \(c^{2}\) from \(a^{2}\):
\(a^{2}-c^{2}=(h^{2}+b^{2}+2bx + x^{2})-(h^{2}+x^{2})\)

Step2: Simplify the right - hand side

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Answer:

Subtract the \(c^{2}\) equation from the \(a^{2}\) equation, resulting in \(a^{2}-c^{2}=b^{2}+2bx\).