Question

Given a cuboid ABCD.EFGH with the length of AB = 5 cm, BC = 5 cm, and AE = 3 cm. Find the value of sinus between line AG and plane BCGF. Help me to find geometry problem, please?

Answer 1

`sinalpha=5/sqrt59`

Explanation:

The angle between a line `l` and a plane `pi` is the angle between the line `l` and its orthogonal projection onto the plane `pi`.
See Fig 1.
`AB` is perpendicular to the plane `BCGF`.
`BG` is the orthogonal projection of `AG` on the plane `BCGF`.
The angle between `AG` and plane `BCGF` is `angleAGB`, as shown in Fig 2.
Please note that `A,G and B` forms a right triangle, and `AG` is the hypotenuse.
Let `angleAGB=alpha`
Use Pythagorean theorem to find `GB and AG`,
Consider `DeltaGCB`,
`GB=sqrt(GC^2+BC^2)=sqrt(3^2+5^2)=sqrt34`
Consider `DeltaAGB`,
`AG=sqrt(BA^2+GB^2)=sqrt(5^2+(sqrt34)^2)=sqrt59`

`=> sinalpha=(BA)/(GA)=5/sqrt59`