Simplify \(\frac{a^2-(b-c)^2}{(a+c)^2-b^2}\) + \(\frac{b^2-(a-c)^2}

1.	Simplify \(\frac{a^2-(b-c)^2}{(a+c)^2-b^2}\) + \(\frac{b^2-(a-c)^2}{(a+b)^2-c^2}\) + \(\frac{c^2-(a-b)^2}{(b+c)^2-a^2}\)A) a + b + cB) \(\frac{1}{a+b+c}\)C) 1 D) 0
Answer» Correct option is (C) 1 \(\frac{a^2-(b-c)^2}{(a+c)^2-b^2}+\frac{b^2-(a-c)^2}{(a+b)^2-c^2}+\frac{c^2-(a-b)^2}{(b+c)^2-a^2}\) \(=\frac{(a-(b-c))(a+b-c)}{(a+c-b)(a+c+b)}+\frac{(b-(a-c))(b+a-c)}{(a+b-c)(a+b+c)}+\frac{(c-(a-b))(c+a-b)}{(b+c-a)(b+c+a)}\) \(=\frac{(a+c-b)(a+b-c)}{(a+c-b)(a+b+c)}+\frac{(b-a+c)(a+b-c)}{(a+b-c)(a+b+c)}+\frac{(c-a+b)(a+c-b)}{(c-a+b)(a+b+c)}\) \(=\frac{a+b-c}{a+b+c}+\frac{b-a+c}{a+b+c}+\frac{a+c-b}{a+b+c}\) \(=\frac{a+b-c+b-a+c+a+c-b}{a+b+c}\) \(=\frac{2a-a+2b-b+2c-c}{a+b+c}\) \(=\frac{a+b+c}{a+b+c}\) = 1 Correct option is C) 1

Simplify \(\frac{a^2-(b-c)^2}{(a+c)^2-b^2}\) + \(\frac{b^2-(a-c)^2}{(a+b)^2-c^2}\) + \(\frac{c^2-(a-b)^2}{(b+c)^2-a^2}\)A) a + b + cB) \(\frac{1}{a+b+c}\)C) 1 D) 0

Answer»

Correct option is (C) 1

\(\frac{a^2-(b-c)^2}{(a+c)^2-b^2}+\frac{b^2-(a-c)^2}{(a+b)^2-c^2}+\frac{c^2-(a-b)^2}{(b+c)^2-a^2}\)

\(=\frac{(a-(b-c))(a+b-c)}{(a+c-b)(a+c+b)}+\frac{(b-(a-c))(b+a-c)}{(a+b-c)(a+b+c)}+\frac{(c-(a-b))(c+a-b)}{(b+c-a)(b+c+a)}\)

\(=\frac{(a+c-b)(a+b-c)}{(a+c-b)(a+b+c)}+\frac{(b-a+c)(a+b-c)}{(a+b-c)(a+b+c)}+\frac{(c-a+b)(a+c-b)}{(c-a+b)(a+b+c)}\)

\(=\frac{a+b-c}{a+b+c}+\frac{b-a+c}{a+b+c}+\frac{a+c-b}{a+b+c}\)

\(=\frac{a+b-c+b-a+c+a+c-b}{a+b+c}\)

\(=\frac{2a-a+2b-b+2c-c}{a+b+c}\)

\(=\frac{a+b+c}{a+b+c}\) = 1

Correct option is C) 1