Geometric transformations: homothety, similarity, inversion and their applications
This domain asks you to produce transformed figures and to justify the transformation (centre, ratio, power). The exam expects the construction, the ratio or power written next to the figure, and an understanding of what the transformation preserves: angles and parallelism in homothety and similarity, angles (but not orientation of the plane) in inversion.
Core rules
- Homothety: centre O and ratio k; O, A and A' are collinear with OA' = k·OA. For k > 0 the image is on the same ray, for k < 0 it is on the opposite ray through O. Parallels stay parallel, angles are preserved, lengths scale by |k|, areas by k² and volumes by |k|³.
- Similarity: a homothety composed with a rotation (or translation). Two circles always have two centres of similitude (external and internal), the key to solving tangent-circle problems.
- Inversion: centre O, power K (K = r² for the circle of inversion). O, P and P' are collinear on the same side and OP·OP' = K. Points on the circle of inversion are fixed; O has no image. A line not passing through O inverts to a circle through O, and a circle through O inverts to a line.
Worked example
Circle of inversion centre O, r = 6 cm, so K = 36 cm². A point P at OP = 9 cm gives OP' = 36/9 = 4 cm on the same ray. A circle passing through O with diameter 8 cm inverts to a straight line perpendicular to that diameter at a distance K/d = 36/8 = 4.5 cm from O; the point of the circle diametrically opposite O inverts to the foot of that perpendicular.
Traps that cost marks
Forgetting the sign of k and mirroring the figure through O without saying so; confusing the linear ratio with the area ratio (k² is what changes the size of a surface in a design); placing the inverse point on the opposite ray; treating the centre of inversion as a fixed point; in similarity, giving only one centre of similitude when the problem needs the internal one to obtain an internal tangency.
Applications in design are legitimate content here: enlargement of a panel by homothety, modular scaling, and inversion used to turn a tangent-line problem into an easier tangent-circle problem.