Symmetry and enlargement in space
Find the mirror image of a point in a plane, work out how a volume grows under an enlargement, and find all the planes of symmetry of a cube and a ball.
- Russia: grade 10
Guide
In the Russian programme, symmetry in a point, a line and a plane is taught in Grade 10 (polyhedra); the symmetry of a ball, homothety (enlargement) in space and the motions of space are in Grade 11, briefly at the basic level and in full at the advanced level.
Points A and A₁ are symmetric in a plane if the plane passes through the midpoint of AA₁ and is perpendicular to it. In coordinates, reflecting in the plane Oxz changes the sign of y only; reflecting in the origin changes all three signs.
Under a homothety with coefficient k, lengths are multiplied by |k|, areas by k² and volumes by |k|³. A plane of symmetry maps the solid onto itself.
Common slips: multiplying a volume by k instead of k³, mixing up the image of a point with its projection, and counting only the planes parallel to the faces of a cube.