Chapter 8.1 — Introduction to Similar Figures

Moving a Shape Doesn't Change What It Is

Three transformations move a figure around without changing its size or shape:

• Flip (reflection) — a figure is reflected across a line, producing a mirror image of the original.
• Rotation — a figure turns around a fixed centre point; every point on the shape stays the same distance from that centre throughout the turn.
• Slide (translation) — a figure moves from one position to another without turning or flipping at all.

Whichever of these three a figure undergoes, it stays congruent to where it started. A shape rotated, reflected, or slid is still exactly the same shape and size — congruence survives every one of these movements, which is exactly why they're called rigid transformations — nothing about the figure's own internal measurements ever genuinely bends, stretches, or changes shape at all in the process.

Matching Parts, in the Right Order

For two triangles, ∆ABC ≅ ∆PQR means every corresponding part matches exactly: AB = PQ, BC = QR, AC = PR, and ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R. The order the vertices are written in isn't cosmetic — it names exactly which vertex corresponds to which. Writing ∆ABC ≅ ∆QPR instead would claim A corresponds to Q and B corresponds to P, a completely different pairing that generally isn't true even if the two triangles genuinely are congruent to each other under the correct correspondence. Getting the vertex order right is just as important as getting the congruence claim right in the first place.

Same Shape, Different Size — Similarity

Two polygons are similar — denoted with the symbol ~ — when their corresponding angles are all equal and their corresponding sides are all in the same proportion. Similar figures share a shape but not necessarily a size.

For two rectangles with sides 3 and 2 against 4.5 and 3: the ratio of lengths (3/4.5 = 2/3) matches the ratio of breadths (2/3) exactly, so the rectangles are similar. For two triangles with corresponding sides 3, 4, 5 against 6, 8, 10: every one of the three ratios (3/6, 4/8, 5/10) reduces to 1/2, so the triangles are similar too. Some families of shapes are automatically similar to every other member of the same family, no matter the size: all squares are similar to each other, all circles are similar to each other, and all equilateral triangles are similar to each other, since each family's defining condition already forces equal angles and proportional sides.

Congruent Figures Are Always Similar — Never the Reverse

Every congruent pair is automatically a similar pair too, since equal corresponding sides are a special case of proportional corresponding sides (a ratio of exactly 1:1). But similarity is the broader category — two similar figures don't have to be the same size at all, so a similar pair isn't automatically congruent. Congruence is the stricter, more specific relationship; similarity is the looser one that congruence always satisfies, which is why it's fair to think of congruent figures as a special, size-matched subset of similar ones.

Why All Squares Are Similar But Not All Rectangles

It's worth being precise about why some shape families are automatically similar to every other member, while others aren't. Every square has four equal sides and four 90° angles — there's only one "shape" a square can be, just at different sizes, so any two squares automatically satisfy both similarity conditions (equal angles, proportional sides). A rectangle only fixes the angles at 90°; the ratio between its length and breadth can vary freely, so two rectangles are similar only when that specific ratio happens to match, not automatically. The same reasoning explains why all circles are similar (a circle has exactly one shape parameter, its radius, and no angle condition to fail) and why all equilateral triangles are similar (fixing all three angles at 60° and all three sides equal removes every degree of freedom except overall size).

Enlarging or Shrinking on Purpose — Dilation

A dilation is the deliberate construction of an enlarged or reduced copy of a figure, and the ratio between the copy's sides and the original's sides is called the scale factor.

• Scale factor greater than 1: the new figure is larger than the original.
• Scale factor less than 1: the new figure is smaller than the original.
• Scale factor equal to 1: the new figure is identical in size to the original — a dilation that doesn't actually change anything.

Every dilation produces a figure similar to the original, since the angles never change and every side scales by the same factor — proportional sides and equal angles, exactly the definition of similarity, guaranteed automatically by the construction itself. This is why a dilation is sometimes the quickest way to demonstrate similarity in a construction task: build the copy as a dilation in the first place, and its similarity to the original is guaranteed before a single ratio is even checked.

Why Vertex Order Encodes the Entire Correspondence

The rule about vertex order in ∆ABC ≅ ∆PQR is worth dwelling on, since it's a common source of quiet errors. Writing the congruence statement is really writing three separate claims at once — A matches P, B matches Q, C matches R — compressed into a single line via position alone. This is exactly why ∆ABC ≅ ∆PRQ is a different (and often false) statement even for the very same two triangles: it claims B matches R and C matches Q, the opposite pairing from before. Two triangles can genuinely be congruent under one correspondence and not another, which is why textbooks are careful to write ∆ABC ≇ ∆QPR or ∆ABC ≇ ∆PRQ explicitly when only one specific vertex order actually holds — the shapes are the same, but the claimed correspondence is not.

Definitions Ready for Direct Use

These definitions are put to direct use in Exercise 8.1, including real measurement and construction tasks built around dilation and similar triangles. The rigid transformations covered here — flips and rotations in particular — reappear in Exercise 8.2, where a figure's own reflection or rotation is compared back to itself rather than to a separate figure. Congruence is explored in far more formal depth, with proof-based criteria, in Triangles in Class 9, and similarity gets the same treatment in Similar Triangles in Class 10, where similarity becomes the basis for proving several major geometric theorems rather than just a comparison between two shapes.