Transformation
Class 9: Optional Math
Transformation, Reflection, Rotation, Translation, Dilation, Example Questions with Answers
Transformation
Transformation refers to the movement of a shape or figure in a coordinate plane without changing its size or shape (except in dilation). There are four main types of transformations: Reflection, Rotation, Translation, and Dilation.
Reflection (Flipping over a line)
A reflection flips a figure over a given axis or line. The formulas depend on the axis of reflection. The formulae are given below:
- Over the x-axis: \( P(x, y) \xrightarrow{\text{Re: x - axis}} P'(x, -y) \)
- Over the y-axis: \( P(x, y) \xrightarrow{\text{Re: y - axis}} P'(-x, y) \)
- Over the line \( (y = x) \): \( P(x, y) \xrightarrow{\text{Re: y = x }} P'(y, x) \)
- Over the line \( (y = -x) \): \( P(x, y) \xrightarrow{\text{Re: y = -x }} P'(-y, -x) \)
- Over the line \( (x = a) \): \( P(x, y) \xrightarrow{\text{Re: x = a }} P'(2a – x, y )\)
- Over the line \( (y = b) \): \( P(x, y) \xrightarrow{\text{Re: y = b }} P'(x, 2b - y )\)
Rotation (Turning around a point)
A rotation rotates a figure counterclockwise about the origin (0, 0) by a given angle:
- 90° counterclockwise: \( P(x, y) \xrightarrow{\text{Ro: 0, +90}} P'(-y, x) \)
- 180° counterclockwise: \( P(x, y) \xrightarrow{\text{Ro: 0, +180}} P'(-x, -y) \)
- 270° counterclockwise: \( P(x, y) \xrightarrow{\text{Ro: 0, +270}} P'(y, -x) \)
- For clockwise rotation, reverse the sign of the angle.
- Note: The rotation for [Ro: 0, 360] will remain the same like \( P(x, y) \xrightarrow{\text{Ro: 0, \pm 360}} p’(x, y) \)
Translation (Sliding without rotating)
A translation moves a figure without rotating or flipping it.
- Formula: \( (x, y) \to (x + a, y + b) \)
- Where \( a \) is the horizontal shift
- \( b \) is the vertical shift
Dilation (Resizing the shape)
A dilation changes the size of a figure but keeps its shape the same. It is defined by a scale factor \( k \):
- Formula: \( (x, y) \to (kx, ky) \)
- If \( k > 1 \), the figure enlarges
- If \( 0 < k < 1 \), the figure shrinks
- If \( k = 1 \), the figure remains the same
