IB MYP 4-5 Maths- Movement on a plane- Study Notes - New Syllabus
IB MYP 4-5 Maths- Movement on a plane – Study Notes
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- Movement on a plane
IB MYP 4-5 Maths- Movement on a plane – Study Notes – All topics
Introduction to Transformations
Introduction to Transformations
A transformation is a change in the position, size, or orientation of a shape in a plane. Transformations in MYP include:
- Translation – moves a shape without rotation or resizing.
- Reflection – flips a shape over a line (mirror line).
- Rotation – turns a shape about a point.
- Enlargement (Dilation) – changes the size of a shape, keeping the shape similar.
Key Properties of Transformations:
- Shape remains congruent (except in enlargement where it remains similar).
- Angles remain the same in all transformations.
- Coordinates change according to specific rules.
Translation
A translation moves a figure without rotating or resizing it. Every point moves the same distance in the same direction.
Representation: A translation is described using a vector \( \begin{pmatrix} a \\ b \end{pmatrix} \), where \(a\) is horizontal movement and \(b\) is vertical movement.
- If \(a > 0\), move right; if \(a < 0\), move left.
- If \(b > 0\), move up; if \(b < 0\), move down.
Example:
Translate point \( P(3, 4) \) by vector \( \begin{pmatrix} -2 \\ 5 \end{pmatrix} \).
▶️Answer/Explanation
Step 1: Translation vector \( (-2, 5) \).
Step 2: Add components to original coordinates:
\( (3 – 2, 4 + 5) = (1, 9) \).
Answer: New coordinates are (1, 9).
Reflection
A reflection flips a shape over a line (called the mirror line). Each point and its image are the same distance from the line of reflection.
Common Lines of Reflection:
- x-axis: \( y = 0 \)
- y-axis: \( x = 0 \)
- Line \( y = x \)
- Line \( y = -x \)
Example:
Reflect point \( (4, -3) \) in the x-axis.
▶️Answer/Explanation
Rule: Reflection in x-axis changes \( (x, y) \) to \( (x, -y) \).
Step: \( (4, -3) \rightarrow (4, 3) \).
Answer: Image is at (4, 3).
Rotation
A rotation turns a figure around a fixed point (the center of rotation) by a given angle in a specific direction (clockwise or anticlockwise).
Common Angles: \(90^\circ\), \(180^\circ\), \(270^\circ\).
About Origin:
- \(90^\circ\) anticlockwise: \( (x, y) \rightarrow (-y, x) \)
- \(180^\circ\): \( (x, y) \rightarrow (-x, -y) \)
- \(270^\circ\) anticlockwise: \( (x, y) \rightarrow (y, -x) \)
Example:
Rotate point \( (3, 2) \) \(90^\circ\) anticlockwise about the origin.
▶️Answer/Explanation
Rule: \( (x, y) \rightarrow (-y, x) \).
Step: \( (3, 2) \rightarrow (-2, 3) \).
Answer: Image is at (-2, 3).
Enlargement (Dilation)
An enlargement changes the size of a figure but keeps the shape similar. It uses a scale factor and a center of enlargement.
Key Notes:
- If \( k > 1 \), image is larger.
- If \( 0 < k < 1 \), image is smaller.
- If \( k = 1 \), No change in image.
- If \( k \) is negative, the image is on the opposite side of the center.
Example:
A point P(4, 6) is enlarged about the origin with scale factor \( \dfrac{1}{2} \). Find its image.
▶️Answer/Explanation
Step: Multiply coordinates by \( \dfrac{1}{2} \):
\( (4, 6) \rightarrow (2, 3) \).
Answer: Image is at (2, 3).
Combination of Transformations
When two or more transformations are applied to a figure in sequence, the result is a combined transformation. The order matters!
Example:
Start with point A(2, 1).
▶️Answer/Explanation
Step 1: Translate by \( (3, 2) \): \( (2+3, 1+2) = (5, 3) \).
Step 2: Reflect in y-axis: \( (5, 3) \rightarrow (-5, 3) \).
Answer: Final image is at (-5, 3).
Tessellations
A tessellation is a pattern formed by repeating a shape (or several shapes) over a plane without any gaps or overlaps. Tessellations are common in art, architecture, and nature.
Key Features of Tessellations:
- A tessellation completely covers the plane without gaps or overlaps.
- Tessellations can use one shape (regular tessellation) or multiple shapes (semi-regular tessellation).
- Transformations used to create tessellations include translation, rotation, and reflection.
Types of Tessellations:
Regular Tessellation:
Uses only one type of regular polygon. There are only 3 possible shapes: equilateral triangles, squares, and regular hexagons.
Semi-Regular Tessellation:
Uses two or more types of polygons arranged in a repeating pattern.
Irregular Tessellations:
Irregular tessellations are composed of shapes that aren’t regular polygons, but they still fit together without leaving any gaps or overlaps.With irregular tessellations, there’s a limitless number of figures you can create.
Condition for Tessellation:
The interior angles of polygons that meet at a point must add up to \(360^\circ\).
Note: To create complex tessellations, shapes can be rotated, reflected, and translated to fit together perfectly.
Example:
Does a regular hexagon tessellate the plane?
▶️Answer/Explanation
Step 1: Interior angle of a regular hexagon = \(120^\circ\).
Step 2: Around a point, 3 hexagons meet: \( 120^\circ + 120^\circ + 120^\circ = 360^\circ \).
Answer: Yes, a regular hexagon tessellates the plane.
Example:
Can a regular pentagon tessellate the plane?
▶️Answer/Explanation
Step 1: Interior angle of a regular pentagon = \(108^\circ\).
Step 2: Check if multiples of \(108^\circ\) can add to \(360^\circ\): \( 360^\circ \div 108^\circ \approx 3.33 \) (not an integer).
Answer: No, a regular pentagon cannot tessellate the plane on its own.