Final answer:
The area of the triangle with corners at P1 = (0, 4, 4), P2 = (4, −4, 4), and P3 = (2, 2, −4) is found to be approximately 28.2843 square units, which rounds to the nearest answer choice as 32 square units. Option B is correct.
Step-by-step explanation:
To find the area of a triangle in Cartesian coordinates with vertices P1 = (0, 4, 4), P2 = (4, −4, 4), and P3 = (2, 2, −4), we can use the vector cross product method. First, we find two vectors that represent two sides of the triangle by subtracting the coordinates of P1 from P2 and P1 from P3.
Vector A = P2 - P1 = (4, −4, 4) - (0, 4, 4) = (4, −8, 0)
Vector B = P3 - P1 = (2, 2, −4) - (0, 4, 4) = (2, −2, −8)
Next, we find the cross product of vectors A and B:
Cross Product = A × B = (|A2B3 - A3B2|, |A3B1 - A1B3|, |A1B2 - A2B1|) = (32, 40, 24)
The area of the triangle is half the magnitude of the cross product:
Area = 0.5 × √(32² + 40² + 24²) = 0.5 × √(1024 + 1600 + 576) = 0.5 × √3200 = 0.5 × 56.5685
Area = 28.2843 square units, which means that the closest answer choice is 32 square units (rounded to the nearest whole number).