Two walls and a ceiling of a room meet at right angles at a point P. A fly is in the air 1 m from one wall, 8 m from the other wall and 9 m from the point P. How many meters is the fly from the ceiling

This is a 3D geometry problem. Let the point P be the origin (0, 0, 0) of a coordinate system. Since the two walls and the ceiling meet at right angles at P, these surfaces can be considered as the coordinate planes (x=0, y=0, z=0). Let the coordinates of the fly be (x, y, z). 1. "1 m from one wall": This means the x-coordinate of the fly is 1 (x = 1). 2. "8 m from the other wall": This means the y-coordinate of the fly is 8 (y = 8). 3. "9 m from the point P": This is the distance from the origin (0,0,0) to the fly (x,y,z). The 3D distance formula is sqrt(x^2 + y^2 + z^2). So, sqrt(1^2 + 8^2 + z^2) = 9. We need to find "how many meters is the fly from the ceiling". If the ceiling is the z=0 plane (as per our coordinate system setup), then this distance is simply the z-coordinate of the fly. Now, let's solve for z: sqrt(1^2 + 8^2 + z^2) = 9 Square both sides: 1^2 + 8^2 + z^2 = 9^2 1 + 64 + z^2 = 81 65 + z^2 = 81 z^2 = 81 - 65 z^2 = 16 z = sqrt(16) z = 4 (Since distance must be positive) So, the fly is 4 meters from the ceiling. Analyzing the options: A) 4: This matches our calculated distance. B) 6: Incorrect. C) 12: Incorrect. D) 15: Incorrect. The final answer is A