Congruence and Similarity: Dilations and Scale Factor

Congruent: same size, same shape

Two figures are congruent when one can be carried exactly onto the other by a sequence of rigid motions — translations, reflections and rotations.

That is a definition you can act on. Rather than judging by eye, you ask whether some sequence of slides, flips and turns lands one figure on the other.

Because every rigid motion preserves lengths and angles, congruent figures have:

• every pair of matching sides equal in length
• every pair of matching angles equal in measure

Dilation: resizing about a center

A dilation is the transformation congruence leaves out. It resizes a figure by a scale factor $k$, measured from a fixed center.

About the origin:

$$(x, \; y) \;\rightarrow\; (kx, \; ky)$$

Every coordinate doubled:

$$(1, 1) \rightarrow (2, 2) \qquad (3, 1) \rightarrow (6, 2) \qquad (1, 4) \rightarrow (2, 8)$$

The base went from $2$ to $4$ and the height from $3$ to $6$. Both doubled, which is what keeps the shape.

Scale factor	Effect
$k > 1$	enlarges
$k = 1$	unchanged
$0 < k < 1$	shrinks

Similar: same shape, any size

Two figures are similar when one can be carried onto the other by rigid motions and a dilation.

So similar figures have:

• every pair of matching angles equal
• every pair of matching sides proportional

Congruence is the special case where $k = 1$. Every congruent pair is similar; most similar pairs are not congruent.

	Angles	Sides
congruent	equal	equal
similar	equal	proportional

Why angles survive and lengths do not

Multiplying every coordinate by $k$ multiplies every length by $k$ — that much is the definition.

Angles are untouched for a reason worth seeing. An angle is fixed by the ratio of the sides around it, not by their sizes. A slope of “up 3 for every 2 across” describes the same steepness whether the run is $2$ or $20$. Scaling both by $k$ leaves $\tfrac{3k}{2k} = \tfrac{3}{2}$, so the direction — and therefore the angle — is exactly as it was.

That is the whole reason similar figures are useful. A scale drawing, a map, a photograph enlargement and an architect’s model all rely on shape surviving a change of size, and all of them are dilations.

Finding a missing side

Matching sides are in a constant ratio, so a proportion does the work.

Two similar rectangles: the first is $4$ by $6$, the second has a matching side of $10$ where the first had $4$.

$$k = \frac{10}{4} = 2.5$$ $$\text{missing side} = 6 \times 2.5 = 15$$

Or as a proportion, which avoids computing $k$ separately:

$$\frac{4}{6} = \frac{10}{x} \quad\Rightarrow\quad 4x = 60 \quad\Rightarrow\quad x = 15$$

Match the sides correctly. The side that pairs with $4$ is $10$, so those go together on the same side of the equation.

Worked examples

Common mistakes

Practice problems