IB MYP 4-5 Maths- Identical representation of transformations- Study Notes - New Syllabus
IB MYP 4-5 Maths- Identical representation of transformations – Study Notes
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- Identical representation of transformations
IB MYP 4-5 Maths- Identical representation of transformations – Study Notes – All topics
Identical Transformations
Identical Transformations
Two different transformations are called identical if they produce the same result when applied to any figure. This often happens when a combination of transformations is equivalent to a single transformation.
Key Rules:
- Two reflections in intersecting lines are identical to a rotation about their point of intersection.
- Two reflections in parallel lines are identical to a translation.
- A rotation of \(180^\circ\) about a point is identical to two reflections in perpendicular lines through that point.
Example:
A triangle is reflected in the line \( y = x \) and then in the line \( y = -x \). Describe the single transformation that is identical to these two reflections.
▶️Answer/Explanation
Step 1: Reflection in \( y = x \) sends \( (x, y) \) to \( (y, x) \).
Step 2: Reflection in \( y = -x \) sends \( (y, x) \) to \( (-x, -y) \).
Step 3: So, \( (x, y) \rightarrow (-x, -y) \), which is a rotation of \(180^\circ\) about the origin.
Answer: The combined transformation is a \(180^\circ\) rotation about the origin.
Example: Show that two reflections in parallel lines 2 cm apart are identical to a translation.
▶️Answer/Explanation
Step 1: Reflect a point across the first line.
Step 2: Reflect the image across the second line. The point moves in a straight line perpendicular to the two lines.
Step 3: The distance moved is twice the distance between the lines: \( 2 \times 2 \text{ cm} = 4 \text{ cm} \).
Answer: The two reflections are identical to a translation of 4 cm perpendicular to the lines.
Invariant Points and Lines
An invariant point or line remains unchanged after a transformation.
Examples:
- In reflection across a line, all points on the line of reflection are invariant.
- In rotation, only the center of rotation is invariant.
- In translation, no point (except in zero translation) is invariant.
- In enlargement about a point, the center of enlargement is invariant.
Example:
What are the invariant points when a triangle is reflected in the line \( y = x \)?
▶️Answer/Explanation
Answer: All points lying on the line \( y = x \) remain unchanged after reflection, so these are the invariant points.
Matrix Representation of Transformations
A transformation in the coordinate plane can be represented by a \( 2 \times 2 \) matrix that multiplies column vectors of coordinates.
Key Matrices:
- Reflection in x-axis: \( \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \)
- Reflection in y-axis: \( \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \)
- Rotation \(90^\circ\) anticlockwise: \( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \)
- Rotation \(180^\circ\): \( \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \)
- Enlargement by \(k\): \( \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} \)
Example:
Apply a rotation of \(90^\circ\) anticlockwise about the origin to point (2, 3) using a matrix.
▶️Answer/Explanation
Step 1: Matrix for \(90^\circ\) anticlockwise is \( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \).
Step 2: Multiply by column vector: \( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} -3 \\ 2 \end{pmatrix} \).
Answer: New coordinates are (-3, 2).
Example:
Point \( (5, -2) \) is reflected in the y-axis using a transformation matrix. Find its image.
▶️Answer/Explanation
Step 1: Matrix for reflection in y-axis: \( \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \)
Step 2: Multiply matrix by the column vector:
\( \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 5 \\ -2 \end{pmatrix} = \begin{pmatrix} (-1)(5) + (0)(-2) \\ (0)(5) + (1)(-2) \end{pmatrix} = \begin{pmatrix} -5 \\ -2 \end{pmatrix} \)
Answer: The image of the point is \( (-5, -2) \).