🔢 Transformations
A translation moves a shape cleanly along a straight track without tilting it
▶ Play Door 13Translation, Reflection, and Rotation
What is a transformation? In geometry, a transformation is a way of moving or changing a geometric shape. The original shape is called the object, and the new shape after the movement is called the image.
Translation: A translation means sliding a shape across a flat surface without turning it, flipping it, or changing its size. Every point of the shape moves by the exact same distance and in the exact same direction. Think of pushing a chess piece forward on a chessboard, or sliding a video game character across the screen. We can describe a translation using a column vector:
\[ pmatrix x \ y pmatrix \]
Where \(x\) tells us how many steps to move horizontally (right is positive, left is negative) and \(y\) tells us how many steps to move vertically (up is positive, down is negative).
Reflection: A reflection is a flip over a straight line called the line of…
Worked example
Problem: A point \(A(3, 5)\) is translated using the vector \(pmatrix 4 \ -2 pmatrix\). Find the coordinates of its image \(A'\).
- *Step 1: Identify the starting coordinates of the object point: \(x = 3, y = 5\).
- *Step 2: Read the translation values from the column vector.
- The top number (\(4\)) means add \(4\) to the \(x\)-coordinate.
- The bottom number (\(-2\)) means subtract \(2\) from the \(y\)-coordinate.
- *Step 3: Calculate the new \(x\)-coordinate:* \(x' = 3 + 4 = 7\)
Answer: *Step 3: Calculate the new \(x\)-coordinate:* \(x' = 3 + 4 = 7\)
Line Symmetry & Rotational Symmetry
What is symmetry? A geometric shape has symmetry if it can be folded, split, or turned in a way that makes it fit perfectly onto itself. It means the shape looks balanced and repeating.
Line Symmetry: A shape possesses line symmetry (or reflectional symmetry) if you can draw a straight line straight through it that splits it into two identical halves. If you fold the shape along this line, the two halves will overlap perfectly with no edges left hanging over. This folding line is called a line of symmetry. Think of a butterfly, a heart symbol, or the human face. Different shapes have different numbers of lines:
• A human hand has \(0\) lines of symmetry.
• A standard rectangle has exactly \(2\) lines of symmetry (running vertically and horizontally through the center).
• A regular square has \(4\) lines of symmetry (vertical, horizontal, and both diagonals).…
Worked example
Problem: Determine the number of lines of symmetry and the order of rotational symmetry for a regular hexagon.
- *Step 1: Recall the geometric rule for all regular polygons (shapes where all sides and internal angles are identical): the number of lines of symmetry matches the number of sides.
- *Step 2: Count the sides of a hexagon (\(6\) sides).
- Therefore, it has \(6\) lines of symmetry (\(3\) passing through opposite corners and \(3\) cutting through opposite flat side midpoints).
- *Step 3: Determine the rotational order.
- A regular polygon also has an order of rotational symmetry equal to its number of sides.
Answer: A regular polygon also has an order of rotational symmetry equal to its number of sides.
Enlargement and Scaling
What is enlargement? An enlargement is a geometric transformation that changes the size of a shape while keeping its proportions and internal angles exactly the same. Despite the name "enlargement," this process can make a shape bigger or smaller! The resulting image is similar to the original object, but not congruent. Think of pinching to zoom into a photograph on a phone screen, or using a projector to display a laptop screen onto a wall.
To perform an enlargement, you must know two pieces of information:
• The center of enlargement: The fixed anchor point from which lines are drawn through all vertices of the shape to expand or shrink it.
• The scale factor (\(k\)): The multiplier value used to alter the side lengths.
How Scale Factors Work:
• If \(k > 1\), the shape grows larger (e.g., \(k = 2\) means the image is twice as large as the original).
• If \(0…
Worked example
Problem: A triangle has a base side length of \(6 cm\). It undergoes an enlargement with a scale factor of \(k = 3\). Calculate the base side length of the resulting enlarged image triangle.
- *Step 1: Identify the given dimensions: Object side length \(= 6 cm\), Scale factor \(k = 3\).
- *Step 2: Recall the core scale factor conversion formula:* \(Image Length = Object Length × k\).
- *Step 3: Substitute the known parameters into the equation:* \(Image Length = 6 cm × 3\)
- *Step 4: Multiply out to get the final
Answer: * \(Image Length = 18 cm\).