Enlargements

• If $k > 1$, the image is larger than the original object and is located further away from the center of enlargement on the same side.

• If $0 < k < 1$, the transformation is a reduction; the image is smaller than the original and is located closer to the center of enlargement.

• If $k$ is negative, the image is projected through the center to the opposite side. This results in an image that is inverted (rotated $180^{\circ}$) relative to the original object.

• The magnitude of $k$ (its absolute value $|k|$) determines the size change, while the sign determines the direction and orientation.

• Linear Scale Factor ($k$): All corresponding lengths (sides, perimeters, diagonals) change by the factor $k$. For example, if $k=3$, every side of the image is 3 times longer than the original.

• Area Scale Factor ($k^2$): The area of the enlarged image is the original area multiplied by the square of the scale factor. This is because area is a two-dimensional measure ($k \times k$).

• Volume Scale Factor ($k^3$): For three-dimensional objects, the volume of the enlarged image is the original volume multiplied by the cube of the scale factor ($k \times k \times k$).

• Angle Invariance: All internal angles of the shape remain unchanged. This property ensures that the object and image are similar, not congruent (unless $|k|=1$).

The Ray Method

• Draw straight lines (rays) from the center of enlargement through each vertex of the original object.
• Measure the distance from the center to a vertex, multiply it by the scale factor, and mark the new point along the same ray.

The Coordinate Method

• Identify the horizontal ($x$) and vertical ($y$) distances from the center of enlargement to a vertex.
• Multiply both distances by the scale factor $k$.
• From the center of enlargement, count out these new distances to locate the corresponding vertex of the image.

Key Formula: $\text{Distance from Center to Image} = k \times \text{Distance from Center to Object}$

• Unlike other transformations like reflections or rotations, enlargement is the only basic transformation that typically alters the area and perimeter of the shape.

• Finding the Center: If an object and its image are given, find the center of enlargement by drawing lines through at least two pairs of corresponding vertices. The point where these lines intersect is the center.

• Calculating Scale Factor: Always use the formula $k = \frac{\text{Image Length}}{\text{Object Length}}$. Ensure you are comparing corresponding sides, not just any two sides.

• Checking Negative Enlargements: Remember that a negative scale factor results in an image on the opposite side of the center. If your image is on the same side as the object, the scale factor must be positive.

• Area Verification: If a question provides areas, remember to take the square root of the area ratio to find the linear scale factor $k$ before applying it to lengths.