3-8 Transformations - Grade 9 Math
Key Concepts
1. Translation
Translation is a transformation that moves every point of a figure the same distance in the same direction. It can be described using a vector or a pair of coordinates.
2. Rotation
Rotation is a transformation that turns a figure around a fixed point, called the center of rotation. The amount of turning is specified by an angle.
3. Reflection
Reflection is a transformation that flips a figure over a line, called the line of reflection. The reflected figure is a mirror image of the original figure.
4. Dilation
Dilation is a transformation that changes the size of a figure without changing its shape. It is described by a center and a scale factor.
Detailed Explanation
Translation
In translation, every point of the figure is moved the same distance and direction. For example, translating a triangle by (3, 2) means moving each vertex of the triangle 3 units to the right and 2 units up.
Example: Translate triangle ABC with vertices A(1, 2), B(3, 4), and C(2, 5) by (2, -1).
New vertices: A'(3, 1), B'(5, 3), C'(4, 4).
Rotation
Rotation involves turning a figure around a center point. For example, rotating a triangle 90° counterclockwise around the origin means each vertex is rotated 90° in the counterclockwise direction.
Example: Rotate triangle ABC with vertices A(1, 2), B(3, 4), and C(2, 5) 90° counterclockwise around the origin.
New vertices: A'(-2, 1), B'(-4, 3), C'(-5, 2).
Reflection
Reflection involves flipping a figure over a line. For example, reflecting a triangle over the y-axis means each vertex is reflected across the y-axis.
Example: Reflect triangle ABC with vertices A(1, 2), B(3, 4), and C(2, 5) over the y-axis.
New vertices: A'(-1, 2), B'(-3, 4), C'(-2, 5).
Dilation
Dilation involves changing the size of a figure. For example, dilating a triangle by a scale factor of 2 means each side of the triangle is doubled in length.
Example: Dilate triangle ABC with vertices A(1, 2), B(3, 4), and C(2, 5) by a scale factor of 2 centered at the origin.
New vertices: A'(2, 4), B'(6, 8), C'(4, 10).
Analogies for Clarity
Translation as a Move
Think of translation as moving a piece of furniture in a room. The furniture retains its shape and size but is placed in a new position.
Rotation as a Spin
Think of rotation as spinning a top. The top retains its shape and size but changes its orientation as it spins.
Reflection as a Mirror
Think of reflection as looking at your reflection in a mirror. Your image retains your shape and size but is flipped horizontally.
Dilation as a Zoom
Think of dilation as zooming in or out on a photograph. The photograph retains its shape but changes in size.