Transform every vertex

1 · Learn

A rotation turns a shape about a fixed centre. State the centre, angle and direction: 90° anticlockwise about O is a different instruction from 90° clockwise about O. Every point stays the same distance from O, so side lengths and angles are preserved. Trace the whole triangle, keep the centre fixed and turn the paper to check.

On equal-scale axes, a 90° anticlockwise rotation about (0,0) maps (x,y) to (-y,x). For triangle A(1,1), B(3,1), C(1,2), the images are A'(-1,1), B'(-1,3), C'(-2,1), as shown. Check all three vertices, not just A. A full turn of 360° returns the original shape.

An enlargement of scale factor k about a centre multiplies each distance from that centre by k along the same ray when k is positive. About (0,0), factor 2 maps (x,y) to (2x,2y). The lower diagram shows the ORIGINAL triangle enlarged to (2,2), (6,2), (2,4), not the already rotated triangle. Lengths double; angles stay equal.

A factor between 0 and 1 makes a smaller similar image. A different centre changes the coordinates: for centre (1,1), doubling P(3,2) doubles its displacement (2,1), giving (1,1)+(4,2)=(5,3). Simply doubling both coordinates would use the wrong centre.

2 · Worked example

Rotate P(2,1) 90° anticlockwise about O: P'=(-1,2). Its squared distance from O is 2²+1²=5 before and (-1)²+2²=5 after. Separately enlarge P by factor 3 about O: P''=(6,3); its distance from O triples, not stays equal.

3 · Your turn

Rotate triangle (1,1), (3,1), (1,2) 90° clockwise about O. Separately enlarge the original triangle by factor 1/2 about O. State what happens to side lengths in each case.

4 · Apply your learning

On squared paper rotate and enlarge the supplied original triangle. Label original and image vertices. Check a horizontal side length and the distance of a vertex from the specified centre. Explain why an enlargement is not usually congruent to its original.