Enlargements

• Distance Proportionality: The distance from the center of enlargement to any point on the image is exactly $k$ times the distance from the center to the corresponding point on the object. This applies to both direct distance and horizontal/vertical components.

• Linearity of Scale: If the scale factor is $k$, every linear dimension (perimeter, side length, diagonal) increases by a factor of $k$. However, the area of the shape increases by a factor of $k^2$.

• Invariance of Angles: Enlargements are conformal transformations, meaning they preserve the angles between lines. This is why the resulting image is always similar to the original.

• Orientation: For positive scale factors, the image remains in the same orientation as the object. For negative scale factors, the image is inverted and appears on the opposite side of the center of enlargement.

Performing an Enlargement

• Step 1: Identify the Center and Scale Factor: Locate the coordinates of the CoE and determine the value of $k$.

• Step 2: Measure Vector Distances: For each vertex of the object, calculate the horizontal ($x$) and vertical ($y$) distance from the CoE. For example, if the CoE is $(1,1)$ and a vertex is $(3,4)$, the distance is $(+2, +3)$.

• Step 3: Apply the Scale Factor: Multiply these distances by $k$. Using the previous example with $k=3$, the new distances become $(+6, +9)$.

• Step 4: Plot the Image Vertices: Starting from the CoE, move the new distances to find the image vertex. In the example, $(1+6, 1+9) = (7, 10)$.

Finding the Center of Enlargement

• To find the CoE when given an object and its image, draw straight lines (rays) connecting at least two pairs of corresponding vertices. The point where these lines intersect is the Center of Enlargement.

• Fractional Scale Factors: When $k$ is a fraction between 0 and 1 (e.g., $1/2$), the image is smaller than the object. This is often called a 'reduction', but mathematically it is still classified as an enlargement.

• Negative Scale Factors: A negative $k$ (e.g., $-2$) means the image is rotated 180 degrees about the center. The distance from the center is still multiplied by the absolute value $|k|$, but in the opposite direction.

• The Ray Line Check: Always draw light pencil lines from the center through the vertices of your object. Your image vertices MUST lie exactly on these lines; if they don't, your calculation is incorrect.

• Verify the Scale Factor: Pick one horizontal or vertical side of the object and the corresponding side of the image. Divide the image length by the object length to ensure it matches the given $k$.

• Coordinate Precision: When counting squares from the center, be careful not to count from the origin $(0,0)$ by mistake. All measurements must start from the Center of Enlargement.

• Area vs. Length: If a question mentions that the area has increased by 9 times, remember that the scale factor for lengths is the square root, which would be $k=3$.