If a, b and c are the sides of a right-angled triangle, where a \gt b \gt c, then what is the value of the expression (a+b+c)(a+b-c)(a-b+c)(a-b-c)?
- A. 4b^{2}c^{2}
- B. -4b^{2}c^{2} ✓
- C. -2a^{2}b^{2}
- D. -4a^{2}b^{2}
Correct Answer: B. -4b^{2}c^{2}
Explanation
Group the terms to apply difference of squares: (a+(b+c))(a-(b+c)) \times (a+(b-c))(a-(b-c)) = (a^2 - (b+c)^2)(a^2 - (b-c)^2). Expanding gives (a^2-b^2-c^2-2bc)(a^2-b^2-c^2+2bc). Since a is the hypotenuse, a^2 = b^2+c^2. Substituting yields (-2bc)(2bc) = -4b^2c^2.
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