If 2s=a+b+c then what is s^{2}+(s-a)(s-b)+(s-b)(s-c)+(s-c)(s-a) equal to?
- A. (a+b+c)^{2}
- B. ab+bc+ca ✓
- C. 2(ab+bc+ca)
- D. 3(ab+bc+ca)
Correct Answer: B. ab+bc+ca
Explanation
Expanding the given expression gives: s^2 + (s^2-as-bs+ab) + (s^2-bs-cs+bc) + (s^2-cs-as+ca) = 4s^2 - 2s(a+b+c) + ab+bc+ca. Substituting a+b+c = 2s, we get 4s^2 - 2s(2s) + ab+bc+ca = 4s^2 - 4s^2 + ab+bc+ca = ab+bc+ca.
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