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Isometry — Definition, Properties & Examples

Isometry

A transformation that is invariant with respect to distance. That is, the distance between any two points in the pre-image must be the same as the distance between the images of the two points.

 

Isometries: Reflections, rotations, translations, glide reflections

Not isometries: Stretches, shrinks, shears

Key Formula

d(P,Q)=d(P′,Q′)d(P, Q) = d(P', Q')
Where:
  • P,QP, Q = Any two points in the pre-image (original figure)
  • P′,Q′P', Q' = The corresponding images of P and Q after the transformation
  • dd = The distance function (Euclidean distance)

Worked Example

Problem: Triangle ABC has vertices A(1, 2), B(5, 2), and C(3, 6). The triangle is reflected over the x-axis to produce triangle A'B'C'. Show that this reflection is an isometry by comparing the side lengths before and after the transformation.
Step 1: Find the image of each vertex under reflection over the x-axis. Reflecting over the x-axis sends (x, y) to (x, −y).
A′(1,−2),B′(5,−2),C′(3,−6)A'(1, -2),\quad B'(5, -2),\quad C'(3, -6)
Step 2: Compute the length of side AB in the pre-image using the distance formula.
AB=(5−1)2+(2−2)2=16=4AB = \sqrt{(5-1)^2 + (2-2)^2} = \sqrt{16} = 4
Step 3: Compute the length of side A'B' in the image.
A′B′=(5−1)2+(−2−(−2))2=16=4A'B' = \sqrt{(5-1)^2 + (-2-(-2))^2} = \sqrt{16} = 4
Step 4: Compute the length of side AC and A'C'.
AC=(3−1)2+(6−2)2=4+16=20AC = \sqrt{(3-1)^2 + (6-2)^2} = \sqrt{4+16} = \sqrt{20}
Step 5: Confirm A'C' matches.
A′C′=(3−1)2+(−6−(−2))2=4+16=20A'C' = \sqrt{(3-1)^2 + (-6-(-2))^2} = \sqrt{4+16} = \sqrt{20}
Step 6: Compute BC and B'C'.
BC=(3−5)2+(6−2)2=4+16=20BC = \sqrt{(3-5)^2 + (6-2)^2} = \sqrt{4+16} = \sqrt{20}
Step 7: Confirm B'C' matches.
B′C′=(3−5)2+(−6−(−2))2=4+16=20B'C' = \sqrt{(3-5)^2 + (-6-(-2))^2} = \sqrt{4+16} = \sqrt{20}
Answer: All three side lengths are preserved: AB = A'B' = 4, AC = A'C' = √20, and BC = B'C' = √20. Because every distance is unchanged, the reflection is an isometry.

Frequently Asked Questions

Is a dilation an isometry?
No. A dilation multiplies all distances by a scale factor. Unless that scale factor is exactly 1 (which leaves the figure unchanged), distances change, so a dilation is not an isometry. For example, a dilation with scale factor 2 doubles every length, making the image larger than the pre-image.
What properties does an isometry preserve besides distance?
Because distances are preserved, isometries also preserve angle measures, perimeter, area, and the overall shape and size of a figure. The pre-image and image are always congruent. Collinearity (whether points lie on the same line) and betweenness of points are preserved as well.

Isometry vs. Dilation

An isometry keeps all distances exactly the same, producing a congruent figure. A dilation multiplies all distances by a constant scale factor k, producing a similar figure. When k = 1, a dilation happens to be an isometry, but for any other value of k, distances change. Think of it this way: isometries preserve both shape and size, while dilations preserve shape but change size.

Why It Matters

Isometries are the foundation of congruence in geometry. Two figures are congruent precisely when one can be mapped onto the other by a sequence of isometries. This connection lets you prove that triangles or other shapes are congruent by identifying the specific reflections, rotations, or translations that carry one onto the other, which is central to geometric proofs and real-world applications like tiling patterns and engineering tolerances.

Common Mistakes

Mistake: Assuming any transformation that keeps a figure 'looking the same shape' is an isometry.
Correction: Preserving shape alone is not enough. A dilation with scale factor 3 keeps the same shape but triples all lengths, so it is not an isometry. An isometry must preserve actual distances, not just proportions.
Mistake: Forgetting that a glide reflection is an isometry.
Correction: A glide reflection combines a reflection with a translation along the line of reflection. Both components are isometries, and their composition is also an isometry. It is the fourth (and often overlooked) type of isometry in the plane.

Related Terms

  • Transformations — Broad category that includes isometries and non-isometries
  • Reflection — An isometry that flips across a line
  • Rotation — An isometry that turns around a fixed point
  • Translation — An isometry that slides every point equally
  • Glide Reflection — An isometry combining reflection and translation
  • Dilation — A non-isometric transformation that scales distances
  • Invariant — Distance is the invariant property of isometries
  • Pre-Image — The original figure before the isometry acts