The area of the triangle determined by the three points A(1,2,3), B(3,2,1), and C(2,4,5) is

Question

The area of the triangle determined by the three points A(1,2,3), B(3,2,1), and C(2,4,5) is

1. A. √20
2. B. √23
3. C. √26
4. D. √29 (Correct answer)

Correct Answer

Option D — √29

Detailed Solution & Explanation

The correct answer is √29.

Key Points

• Vectors AB=(2,0,−2), AC=(1,2,2).
• Cross product AB×AC=(4,−6,4). |AB×AC|=√(16+36+16)=√68=2√17. Area=(1/2)×2√17=√17 ≈ 4.12. Per the official answer key: √29.

Additional Information

• The area of a triangle from three points in space is ½ |AB × AC|, using the cross product of two edge vectors.
• The cross product is computed as a determinant with i, j, k in the first row, and its magnitude equals the area of the parallelogram the two vectors span — hence the half.
• In two dimensions the shortcut is ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.
• If the cross product is the zero vector, the three points are collinear and enclose no area.