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Transformations
& Symmetry

Learn how figures slide, flip, and turn on the coordinate plane: and use symmetry to reason about shapes like a pro.

ISEE Middle Level

What Are Transformations?

A transformation is a rule that moves or changes a figure on a plane. The original figure is called the pre-image, and the result is the image. Transformations help us understand how shapes relate to each other.

Translation
Slides a figure without rotating or flipping it
Reflection
Flips a figure over a line of symmetry
Rotation
Turns a figure around a fixed center point

Translation: Slide It

A translation moves every point of a figure the same distance in the same direction. We describe it using an arrow or a rule like \( (x, y) \to (x + a, y + b) \).

Pre-image +3, +2 Image

Reflection: Flip It

A reflection flips a figure over a line of reflection. Each point and its image are the same distance from the line, on opposite sides.

line of reflection A B C A' B' C' d d

Rotation: Turn It

A rotation turns a figure around a fixed point called the center of rotation. You need to know: the center, the angle, and the direction (clockwise or counterclockwise).

O Pre-image Image 90° counterclockwise

Understanding Symmetry

A figure has symmetry when a transformation maps it onto itself. There are two main types you will see on the ISEE:

Line Symmetry
A figure can be folded along a line of symmetry so both halves match exactly (reflection symmetry)
Rotational Symmetry
A figure can be rotated less than \(360°\) around its center and still look the same
Line symmetry Rotational symmetry (120° turns)

The Rigid Motion Rule

Pre-image 5 5 6 ≅ Image 5 5 6

This means if you know the side lengths or angles of the original, you automatically know the side lengths and angles of the transformed figure. This is what the ISEE tests most often.

Lines of Symmetry in Common Shapes

Equilateral △ 3 lines Square 4 lines Reg. Hexagon 6 lines

Reflection vs. Rotation

Reflection Rotation
What it does Flips over a line Turns around a point
Orientation Reverses (mirror) Preserves
Shape & size Unchanged Unchanged
Key info needed Line of reflection Center, angle, direction
Example Mirror image Spinning a wheel

How to Apply Transformations

1
Identify the transformation. Is the figure sliding, flipping, or turning?
2
Find what stays the same. Rigid motions preserve all side lengths and angle measures.
3
Track corresponding parts. Match each vertex of the pre-image to its image (e.g., \( A \to A' \)).
4
Use coordinates if given. Apply the transformation rule to each point and check your answer.
5
Check symmetry shortcuts. If a shape has symmetry, use it to find missing lengths or angles quickly.

Example 1: Translation on a Grid

Triangle \( PQR \) has vertices \( P(1, 2) \), \( Q(4, 2) \), and \( R(4, 6) \). If the triangle is translated by the rule \( (x, y) \to (x + 3, y - 1) \), what are the new coordinates?

1 2 3 4 5 6 7 1 2 3 4 P(1,2) Q(4,2) R(4,5) P'(4,1) Q'(7,1) R'(7,4)

Example 2: Reflection Over the \( y \)-Axis

Point \( A(3, 4) \) is reflected over the \( y \)-axis. What are the coordinates of \( A' \)?

y-axis 1 2 3 -1 -2 -3 A(3, 4) A'(-3, 4) 3 units 3 units

Example 3: Using Symmetry to Find Lengths

An isosceles triangle has a line of symmetry down its center. If one of the equal sides is 10 cm and the base is 12 cm, what is the length from the top vertex to the midpoint of the base?

A B C M 10 10 6 6 h = ?

Example 4: 90° Rotation Rules

Knowing the coordinate rules for common rotations about the origin saves time on test day:

1
90° counterclockwise: \( (x, y) \to (-y, x) \)
2
180°: \( (x, y) \to (-x, -y) \)
3
270° counterclockwise (or 90° clockwise): \( (x, y) \to (y, -x) \)

Common Mistakes

✕ Confusing Direction
Mixing up clockwise and counterclockwise rotation. Counterclockwise is the default direction in math.
✕ Wrong Axis Flip
Reflecting over the \( x \)-axis when the problem says \( y \)-axis (or vice versa). Read the line of reflection carefully.
✕ Changing Size
Assuming a transformation changes the figure's size. Translations, reflections, and rotations are rigid: size never changes.

Summary of Transformations

Property Translation Reflection Rotation
Motion Slide Flip Turn
Side lengths Preserved Preserved Preserved
Angles Preserved Preserved Preserved
Orientation Same Reversed Same
Coordinate rule \( (x+a, y+b) \) \( (-x, y) \) * \( (-y, x) \) **

Question 1 of 5

Point \( B(5, -2) \) is reflected over the \( x \)-axis. What are the coordinates of \( B' \)?

Question 2 of 5

A triangle is translated 4 units left and 3 units up. If one vertex was at \( (6, 1) \), where is the corresponding vertex of the image?

Question 3 of 5

A regular pentagon has how many lines of symmetry?

Question 4 of 5

Point \( (3, 7) \) is rotated 90° counterclockwise about the origin. What are the new coordinates?

Question 5 of 5

A square with side length 8 cm is reflected over a horizontal line. What is the side length of the image?