Lessons Grade 7

Chapter 19

Chapter 19: Transformations

19.1 Types of Transformations

A transformation is a way of changing a shape’s position on the coordinate plane while maintaining some or all of its properties. Transformations are widely used in geometry, computer graphics, architecture, and physics.

There are four main types of transformations:

Translation (Sliding)

Moves a shape left, right, up, or down without changing its size, shape, or orientation.

Every point in the shape moves the same distance in the same direction.

Example: If a point at (2,3) is translated 4 units right and 2 units up, the new point is (6,5).

Rotation (Turning)

Spins a shape around a fixed point (often the origin) by a certain number of degrees (90°, 180°, etc.).

The shape’s size and shape remain unchanged, but its orientation changes.

Example: Rotating (2,3) 90° counterclockwise around (0,0) moves it to (-3,2).

Reflection (Flipping)

Creates a mirror image of a shape across a line of reflection (like the X-axis or Y-axis).

Each point on the shape is the same distance from the line of reflection, but on the opposite side.

Example: Reflecting (4,2) over the Y-axis results in (-4,2).

Dilation (Resizing)

Changes the size of a shape while keeping its proportions the same.

Uses a scale factor to determine the new size.

Example: If a shape is dilated by a scale factor of 2, every coordinate doubles. (3,4) → (6,8).

19.2 Properties of Transformations

Translations, rotations, and reflections are known as rigid transformations because they do not change the size or shape of the object.

Dilation is a non-rigid transformation because it changes the size but keeps the shape proportional.

Reflections and rotations affect the orientation of a shape, while translations and dilations do not.

Special Rotation Rules Around (0,0):

90° Counterclockwise: (x,y)→(−y,x)(x, y) to (-y, x)(x,y)→(−y,x)

180° Counterclockwise: (x,y)→(−x,−y)(x, y) to (-x, -y)(x,y)→(−x,−y)

270° Counterclockwise: (x,y)→(y,−x)(x, y) to (y, -x)(x,y)→(y,−x)

Reflection Rules:

Over X-axis: (x,y)→(x,−y)(x, y) to (x, -y)(x,y)→(x,−y)

Over Y-axis: (x,y)→(−x,y)(x, y) to (-x, y)(x,y)→(−x,y)

Over y = x: (x,y)→(y,x)(x, y) to (y, x)(x,y)→(y,x)

19.3 Real-World Applications of Transformations

Transformations are used in many real-life situations, such as:

✅ Computer Graphics & Animation – Video game characters use translations to move and rotations for realistic motion.

✅ Architecture & Engineering – Reflections and rotations are used in symmetrical building designs.

✅ Robotics & AI – Robots use translations and rotations to move and position objects.

✅ Medical Imaging – Dilation is used to enlarge or shrink images in X-rays and MRIs.

✅ Fashion Design & Art – Patterns use reflections and translations to create symmetry.

19.4 Exercises

Translate the point (-2,4) 5 units right and 3 units down.

Rotate the point (3,5) 180° around the origin.

Reflect the triangle A(1,2), B(3,4), C(5,6) over the X-axis.

Dilate the shape with points (2,3), (4,5), and (6,7) by a scale factor of 3.

A logo is reflected over the Y-axis and then rotated 90° counterclockwise. How does its position change?