IB MYP 3 Mathematics 8.1 Translations, Rotations and Reflections Study Notes - New Syllabus
IB MYP 3 Mathematics 8.1 Translations, Rotations and Reflections Study Notes
IB MYP 3 Mathematics 8.1 Translations, Rotations and Reflections Study Notes at IITian Academy focus on specific topics and types of questions asked in the actual exam. Study Notes focus on the IB MYP 3 Mathematics syllabus with guiding questions of
Transformation: A movement or change made to a shape that changes its position, orientation, or size according to a specific rule.
Rigid transformation: A transformation that does not change the size or shape of a figure. Translations, rotations and reflections preserve lengths, angles, perimeter and area.
Translation: A transformation that slides every point of a figure the same distance in the same direction.
Translation vector: A vector \(\begin{pmatrix}a\\b\end{pmatrix}\) describing the horizontal and vertical movement of a translated figure.
Rotation: A transformation that turns a figure around a fixed point called the centre of rotation.
Reflection: A transformation that flips a figure across a line called the line of reflection.
Image: The transformed figure produced from the original pre-image.
Pre-image: The original figure before a transformation is applied.
Corresponding points: Points on the pre-image and image that match, commonly labelled using prime notation such as \(A\) and \(A’\).
Key rule: Translation requires a vector, rotation requires a centre, angle and direction, and reflection requires a line of reflection.
8.1 – Translations, Rotations and Reflections
Transformations are movements or changes made to a shape. A transformation changes the position, orientation, or size of a figure according to a specific rule.
In this topic, we study three important transformations:
- Translation – slides a shape
- Rotation – turns a shape around a fixed point
- Reflection – flips a shape across a mirror line
Translations, rotations and reflections are rigid transformations.
They do not change the size or shape of the original figure.
Therefore, corresponding:
• side lengths remain equal
• angle measures remain equal
• perimeter remains unchanged
• area remains unchanged
The transformed figure is called the image, while the original figure is called the pre-image.
\(\text{Pre-image} \longrightarrow \text{Transformation} \longrightarrow \text{Image}\)
Corresponding points are usually labelled using prime notation. For example, \(A\) may move to \(A’\), \(B\) to \(B’\), and so on.
Translation
A translation moves every point of a figure the same distance in the same direction.
A translation is often described using a translation vector.
Translation Vector
A translation vector is written as:
\(\begin{pmatrix} a \\ b \end{pmatrix}\)
where:
\(a\) represents the horizontal movement.
\(b\) represents the vertical movement.
Positive \(a\) → move right
Negative \(a\) → move left
Positive \(b\) → move up
Negative \(b\) → move down
For example, the vector
\(\begin{pmatrix} 4 \\ 2 \end{pmatrix}\)
means move the figure \(4\) units to the right and \(2\) units upwards.
The vector
\(\begin{pmatrix} -3 \\ -5 \end{pmatrix}\)
means move the figure \(3\) units to the left and \(5\) units downwards.
Example of a coordinate translation:
If \(A(2,3)\) is translated by
\(\begin{pmatrix} 4 \\ -2 \end{pmatrix}\)
then:
\(x: 2+4=6\)
\(y: 3-2=1\)
\(A'(6,1)\)
During a translation, every point moves the same distance in the same direction. The orientation of the shape does not change.
Rotation
A rotation turns a figure around a fixed point called the centre of rotation.
A rotation is described by three pieces of information:
- The centre of rotation
- The angle of rotation
- The direction of rotation
The two main directions are:
| Direction | Meaning |
|---|---|
| Clockwise | Turns in the same direction as the hands of a clock |
| Anticlockwise | Turns in the opposite direction to the hands of a clock |
Common angles of rotation include:
| Rotation | Equivalent Rotation |
|---|---|
| \(90^\circ\) clockwise | \(270^\circ\) anticlockwise |
| \(90^\circ\) anticlockwise | \(270^\circ\) clockwise |
| \(180^\circ\) | Same in either direction |
| \(360^\circ\) | Returns to the original position |
When rotating a shape, each point stays the same distance from the centre of rotation.
📌 Important Properties of Rotation
• The shape does not change size.
• The shape does not change its shape.
• Corresponding side lengths remain equal.
• Corresponding angles remain equal.
• Every point rotates through the same angle.
• The distance of each point from the centre of rotation remains unchanged.
Reflection
A reflection flips a figure over a line called the line of reflection.
The line of reflection acts like a mirror. Each point and its image are the same perpendicular distance from the mirror line.
📌 Properties of Reflection
• The size of the figure stays the same.
• The shape stays the same.
• Corresponding side lengths remain equal.
• Corresponding angles remain equal.
• The orientation of the figure is reversed.
• Corresponding points are equally far from the line of reflection.
Common lines of reflection on a coordinate plane include the:
- \(x\)-axis
- \(y\)-axis
- line \(y=x\)
- line \(y=-x\)
For example, reflecting a point across the \(x\)-axis keeps its \(x\)-coordinate unchanged but changes the sign of its \(y\)-coordinate.
\((x,y)\rightarrow(x,-y)\)
Reflecting a point across the \(y\)-axis changes the sign of its \(x\)-coordinate while keeping the \(y\)-coordinate unchanged.
\((x,y)\rightarrow(-x,y)\)
| Transformation | Main Idea | What Happens to the Shape? |
|---|---|---|
| Translation | Slide | Moves the same distance and direction |
| Rotation | Turn | Turns around a fixed centre |
| Reflection | Flip | Flips across a mirror line |
Comparing the Three Transformations
| Feature | Translation | Rotation | Reflection |
|---|---|---|---|
| Changes size? | No | No | No |
| Changes shape? | No | No | No |
| Direction/orientation changes? | No | Yes | Yes |
| Special information needed | Translation vector | Centre, angle and direction | Line of reflection |
How to Describe a Transformation
When asked to describe how one shape has changed into another, give enough information to identify the transformation completely.
| Transformation | What to State |
|---|---|
| Translation | The translation vector |
| Rotation | Centre, angle and direction |
| Reflection | The line of reflection |
Example 1:
Triangle \(ABC\) has vertices \(A(1,2)\), \(B(4,2)\) and \(C(2,5)\). The triangle is translated by
\(\begin{pmatrix} 3 \\ -4 \end{pmatrix}\)
Find the coordinates of \(A’\), \(B’\) and \(C’\).
▶️ Answer/Explanation
Translation: \(3\) units right and \(4\) units down.
Point \(A\):
\(A(1,2)\rightarrow A'(1+3,2-4)\)
\(A'(4,-2)\)
Point \(B\):
\(B(4,2)\rightarrow B'(4+3,2-4)\)
\(B'(7,-2)\)
Point \(C\):
\(C(2,5)\rightarrow C'(2+3,5-4)\)
\(C'(5,1)\)
Final Answer:
\(A'(4,-2),\ B'(7,-2),\ C'(5,1)\)
Example 2:
Point \(P(3,1)\) is transformed in two different ways.
a) Rotate \(P\) \(90^\circ\) anticlockwise about the origin.
b) Reflect the original point \(P\) in the \(x\)-axis.
▶️ Answer/Explanation
a) \(90^\circ\) anticlockwise rotation
For a \(90^\circ\) anticlockwise rotation about the origin:
\((x,y)\rightarrow(-y,x)\)
\(P(3,1)\rightarrow P'(-1,3)\)
Answer: \(P'(-1,3)\)
b) Reflection in the \(x\)-axis
For a reflection in the \(x\)-axis:
\((x,y)\rightarrow(x,-y)\)
\(P(3,1)\rightarrow P'(3,-1)\)
Answer: \(P'(3,-1)\)
Notice that both transformations preserve the distance from the origin, but they move the point in different ways.