Coordinates of a point in space
Decide where a point lies, then find the distance between two points, the midpoint of a segment and a symmetric point from their coordinates.
- France: grade 11
- Ukraine: grade 10
- Russia: grade 11
Guide
Topic from the Ukrainian Grade 10 textbook (algebra and calculus with geometry), section 6 "Coordinates and vectors in space", item 38 "Cartesian coordinates of a point in space". A rectangular coordinate system in space is three pairwise perpendicular number lines through one origin O: the x-axis (abscissa axis), the y-axis (ordinate axis) and the z-axis (applicate axis). The planes through pairs of axes are the coordinate planes xy, xz and yz. Every point M gets an ordered triple (x; y; z), written M (x; y; z).
Distance theorem: the distance between A (x₁; y₁; z₁) and B (x₂; y₂; z₂) is AB = √((x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2). Midpoint theorem: the midpoint of AB has coordinates equal to the halves of the sums of the matching coordinates of the ends. M (x; y; z) and N (x; y; −z) are symmetric about the plane xy; A (x; y; z) and B (−x; −y; −z) are symmetric about the origin.
The figure on the screen is only a box that shows how the coordinates of a point fix its position; the child types the answers.