Geometry

Similarity & Dilations

Two figures are similar if they have the same shape but not necessarily the same size. A dilation is the transformation that produces similar figures — it scales everything by a constant factor while preserving all angles.

Scaling a Function

The simplest way to see dilation in action is to scale a function. Below, the original parabola y = x² is shown alongside a dilated version scaled by factor k.

Scale factor (k)1

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\text{Original: } y = x^2 \qquad \text{Dilated: } y = (1 \cdot x)^2 / 1

Try This

Try this: Set k = 1 and the curves overlap perfectly. Increase k and the red curve gets narrower and taller. Decrease k and it gets wider and shorter. The shape character is preserved — it is still a parabola — but the size changes.

Similar Triangles on a Graph

Two triangles are similar if their corresponding angles are equal (AA similarity). We can show this with lines: the original triangle is formed by lines through the origin, and the dilated triangle is scaled by factor k.

Scale factor1.5

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Both lines pass through the origin, but the dilated line has a shallower slope (2/k). The triangle formed by cutting these with a vertical line will be similar to the original — same angles, proportional sides.

Ratios Stay Constant

The defining property of similar figures: corresponding side ratios are equal. If you scale a triangle by factor k, every side gets multiplied by k, but the ratios between sides stay the same.

Side a3

Side b4

Scale factor2

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\text{Original sides: } a = 3, \; b = 4

\text{Dilated sides: } a' = 3 \times 2, \; b' = 4 \times 2

\frac00 = \frac{ 3 }{ 4 } = \frac{a'}{b'}

No matter what scale factor you pick, the ratio a/b equals the ratio a’/b’. That is similarity in a nutshell.

Connection

Real-world connection: Maps use similarity. A map is a dilation of the real world with a very small scale factor (like 1:50,000). All the angles are preserved, all the distance ratios are preserved — that is why maps work for navigation.

AA, SAS, and SSS Similarity

You do not need to check every side and angle. There are shortcuts:

AA (Angle-Angle): If two angles match, the triangles are similar (the third angle is forced by the 180° rule).
SAS (Side-Angle-Side): If two sides are proportional and the included angle is equal, the triangles are similar.
SSS (Side-Side-Side): If all three pairs of sides are proportional, the triangles are similar.

Challenge

Challenge: Triangle ABC has sides 5, 12, 13. Triangle DEF has sides 10, 24, 26. Are these triangles similar? What is the scale factor? (Hint: check if all ratios are equal.)

Take the Quiz