back to Solid Geometry video lessons
Find the angle between the plane of a triangle and the longest diagonal in a cube
Question:
EFGH-ABCD is a cube. HA is a diagonal in the plane HDAE. AC is a diagonal in the plane ABCD. CH is a diagonal in the plane CGHD. FD is the longest
diagonal in the cube. What is the angle between the plane of the triangle HAC and the line segment FD?
Solution:
- Connect ED

- Because EF is perpendicular to plane EHDA, and ED is on the plane EHDA, so EF is perpendicular to ED.
- So ED is the projection of inclined line segment FD on the plane EHDA.
- Because EHDA is a square, so ED is perpendicular to HA. (This is a property of a square.)
- Because the projection ED is perpendicular to HA, so the inclined line segment FD is also perpendicular to HA (This is a theorem of three perpendicular lines).
- Connect DB

- Because FB is perpendiculat plane BCDA, and DB is on the plane BCDA, so FB is perpendicular to DB.
- So, DB is the projection of FD on the plane BCDA.
- Because BCDA is a square, so DB is perpendicular to AC. (This is a property of a square.)
- Because DB is the projection of FD on the plane BCDA, so FD is perpendicular to AC; (This is a theorem of three perpendicular lines).
- Because HA and AC are two intersected lines and two intersected lines make a plane, so FD is perpendicular to the plane determined by the two intersected line segments, HA and AC.
- So, FD is perpendicular to plane HAC.