How can you tell which figure is the pre-image and which is the image?

The pre-image uses unprimed labels (A, B, C) and represents the original figure before the transformation. The image uses primed labels (A', B', C') and represents the figure after the transformation. If no labels are given, the problem will specify which figure is the starting shape.

Pre-Image — Definition, Formula & Examples

Pre-image is the original figure or point before a transformation is applied. When you reflect, rotate, translate, or dilate a shape, the starting shape is the pre-image and the result is called the image.

Given a transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ , the pre-image of a point $P'$ is the point $P$ such that $T(P) = P'$ . More generally, for any set $S'$ in the range, the pre-image $T^{-1}(S')$ is the set of all points that map to $S'$ under $T$ . In standard geometry notation, original vertices are labeled without primes (e.g., $A, B, C$ ), while their images are labeled with primes ( $A', B', C'$ ).

Key Formula

T(P) = P'

Where:

$T$ = The transformation being applied (reflection, rotation, translation, or dilation)
$P$ = The pre-image point (original point before transformation)
$P'$ = The image point (result after transformation)

How It Works

Whenever you perform a transformation — a reflection, rotation, translation, or dilation — you start with a pre-image and produce an image. To identify the pre-image, look for the original coordinates or the unprimed labels. For example, if triangle

ABC

is reflected over the

y

-axis to produce triangle

A'B'C'

, then

\triangle ABC

is the pre-image and

\triangle A'B'C'

is the image. Under rigid motions (reflections, rotations, translations), the pre-image and image are congruent. Under dilations, they are similar but generally not congruent.

Worked Example

Problem: Triangle ABC has vertices A(2, 3), B(5, 3), and C(5, 7). A reflection over the y-axis is applied. Identify the pre-image and find the image coordinates.

Step 1: Identify the pre-image. The original triangle before the transformation is the pre-image.

\text{Pre-image: } A(2,3),\; B(5,3),\; C(5,7)

Step 2: Apply the reflection over the y-axis. This transformation negates the x-coordinate of each point:

(x, y) \to (-x, y)

A(2,3) \to A'(-2,3)

Step 3: Apply the same rule to the remaining vertices.

B(5,3) \to B'(-5,3), \quad C(5,7) \to C'(-5,7)

Step 4: State the image. The transformed triangle is the image.

\text{Image: } A'(-2,3),\; B'(-5,3),\; C'(-5,7)

Answer: The pre-image is triangle ABC with vertices (2, 3), (5, 3), (5, 7). The image is triangle A'B'C' with vertices (−2, 3), (−5, 3), (−5, 7).

Another Example

Problem: Point P'(6, 10) is the image of point P under a dilation centered at the origin with scale factor 2. Find the pre-image P.

Step 1: Write the dilation rule. A dilation centered at the origin with scale factor

k

maps

(x, y) \to (kx, ky)

T(x, y) = (2x, 2y)

Step 2: To find the pre-image, reverse the transformation by dividing the image coordinates by the scale factor.

P = \left(\frac{6}{2},\; \frac{10}{2}\right) = (3, 5)

Answer: The pre-image is

P(3, 5)

Visualization

Why It Matters

Understanding pre-images is essential in high school geometry courses, especially in units on congruence and similarity where you must describe sequences of transformations. Standardized tests like the SAT and ACT frequently ask you to identify pre-image coordinates or determine which transformation maps a pre-image to its image. In computer graphics and engineering, tracking how original shapes map to transformed versions is fundamental to animation and design.

Common Mistakes

Mistake: Confusing the pre-image with the image, especially when working backward from a transformed figure.

Correction: Remember that the pre-image is always the starting figure (unprimed labels). If a problem gives you the image and asks for the pre-image, you need to reverse the transformation — for example, divide by the scale factor for dilations or reflect back across the same line.

Mistake: Assuming the pre-image and image are always congruent.

Correction: Pre-image and image are congruent only under rigid motions (reflections, rotations, translations). Under dilations with a scale factor other than 1 or −1, the image changes size, so the figures are similar but not congruent.