A hall has a square floor of dimension 10 m × 10 m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is cos ⁻¹1/5, then the height of the hall (in meters) is :

Solution

Maths sir · 4 steps

Put the hall on axes, write AG and BH as vectors, and let the dot product give you the height.

1. Set up axes. Take A as the origin with the floor in the xy-plane: A(0,0,0), B(10,0,0), C(10,10,0), D(0,10,0). Let the height be h, so G (above C) is (10,10,h) and H (above D) is (0,10,h).
2. Diagonal vectors. AG⃗ = (10,10,h) points towards G and BH⃗ = (-10,10,h) points towards H. These are the directions PG and PH, so they make the angle GPH. Their dot product is -100 + 100 + h² = h², and each has squared length 200 + h².

   cos∠ GPH = h²/(200 + h²) = 1/5

3. Solve for h. Cross-multiply: 5h² = 200 + h², so 4h² = 200 and h² = 50.
4. So. h = √50 = 5√2 m, which is option (d).

The idea to keep. For any angle between lines inside a box, place one corner at the origin, read off the coordinates, and use cos θ = (u⃗·v⃗)/|u⃗||v⃗|. No need to chase triangles.

Answer from the NTA final answer key. Solution by PrepWiser.