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?

Correct Option (A)

The scenario describes a three-dimensional geometric problem. Point P, where two walls and a ceiling meet at right angles, can be considered the origin (0,0,0) of a Cartesian coordinate system. The fly's position in the air can be represented by coordinates (x, y, z).

• The distance of the fly from one wall is 1 m. This can be assigned to the x-coordinate, so x = 1 m.
• The distance of the fly from the other wall is 8 m. This can be assigned to the y-coordinate, so y = 8 m.
• The distance of the fly from point P is 9 m. This is the space diagonal from the origin to the fly's position, calculated using the 3D distance formula: √(x² + y² + z²).
• The distance of the fly from the ceiling is the remaining coordinate, z.

Applying the 3D distance formula:

x² + y² + z² = (Distance from P)²

1² + 8² + z² = 9²

1 + 64 + z² = 81

65 + z² = 81

z² = 81 − 65

z² = 16

z = 4 (Since distance must be a positive value)

Therefore, the fly is 4 meters from the ceiling.

Incorrect Options:

Options B (6), C (12), and D (15) are incorrect. Substituting any of these values for 'z' in the 3D distance equation (1² + 8² + z² = 9²) would not satisfy the given condition that the fly is 9 meters from point P. For instance, if z=6, then 1² + 8² + 6² = 1 + 64 + 36 = 101, which is not equal to 9² (81).