Coordinate Transformations

Activities to try

1. 1

   Reflect, then try to copy it with a rotation

   Why it works: Reflection reverses sense while rotation and translation preserve it; this handedness test is what distinguishes a reflection from a rotation even when the image lands in a similar place. Reflecting in y=x swaps the x and y coordinates (reflection in y=−x swaps and negates them: (x,y)→(−y,−x)), not merely slides the shape across.

   1. On a four-quadrant coordinate grid, take a scalene, obviously-handed shape (an 'F' or a flag) and reflect it in the line y=x, recording the image coordinates of each vertex (e.g. (1,4) maps to (4,1) — for y=x, swap the coordinates).
   2. For each vertex not lying on the mirror line, draw the segment from the point to its image and check it meets the mirror line at a right angle, crossing at the midpoint — confirming the mirror is the perpendicular bisector of every such point→image segment (a vertex on y=x maps to itself, so it has no segment).
   3. Note whether the shape's handedness has changed: for the reflection the 'F' should appear mirror-imaged.
   4. Now try to land the ORIGINAL shape on top of your reflected image using a ROTATION instead: pin a centre, choose 90° clockwise, and compare. Observe that the rotated shape keeps its handedness and so cannot coincide with the flipped reflection.
   5. State in your own words why reflection and rotation give different results here: one reverses sense (handedness), the other preserves it.
2. 2

   Fully describe a rotation — centre, angle and direction

   Why it works: A rotation is not fully described — and cannot be uniquely reproduced — without its centre, angle and direction (direction can only be left out for a 180° turn, where both directions give the same image). The centre genuinely changes where the image lands, and can be found by construction.

   1. Rotate the original shape 90° clockwise about the point (0,0) and describe exactly what you did, naming the centre, the angle AND the direction (clockwise/anticlockwise).
   2. Check that your description is complete enough for someone else to reproduce the image uniquely: 'a rotation of 90° clockwise about the origin'.
   3. Carry out two 90° rotations of the same shape about two different centres and compare where the image lands, to feel why the centre is essential and not optional.
   4. Given only an object and its image, reconstruct the centre of rotation by joining two or more object points to their images and intersecting the perpendicular bisectors of those segments.
3. 3

   Translate and prove congruence

   Why it works: All three transformations are rigid motions: they preserve length and angle, so the image is always congruent to the object. Congruence should be checked, not taken on trust. Coordinates are written x before y.

   1. Translate the original shape by the column vector with top entry 4 and bottom entry −3, i.e. 4 right, 3 down (shift every vertex +4 in x and −3 in y), writing the image coordinates.
   2. State whether the lengths and angles have changed, and explain how you know: because the transformation is rigid, the image is congruent to the object.
   3. Prove congruence rather than assume it — trace the object on tracing paper and lay it over the image, or measure corresponding sides and angles, confirming they match.
   4. Record every coordinate as x before y ('along the corridor then up the stairs') to avoid reversing the pair.