What is the difference between an enlargement with $k = 2$ and $k = 0.5$?

A scale factor of $k = 2$ is an expansion, making the image twice as large and further from the center. A scale factor of $k = 0.5$ is a reduction, making the image half the size and closer to the center.

How does a negative scale factor affect the orientation of a shape?

A negative scale factor inverts the shape, effectively rotating it $180^\circ$ around the center of enlargement. The image also appears on the opposite side of the center compared to the original object.

When finding the center of enlargement, why should you use more than one pair of vertices?

While one pair of vertices defines a line the center must lie on, you need at least two lines to find a specific intersection point. Using three pairs acts as a check to ensure no errors were made in identifying corresponding points.

What is a common mistake when calculating the area of an enlarged shape?

Students often multiply the original area by the scale factor $k$ instead of $k^2$. Because enlargement affects two dimensions (length and width), the area increases by the square of the scale factor.

What happens if a student measures distances from the origin $(0,0)$ instead of the specified center of enlargement?

The resulting image will have the correct size and orientation, but it will be in the wrong position on the coordinate plane. The center of enlargement is the only point that remains fixed during the transformation.

Why might an image appear 'inside' the original object during an enlargement?

This occurs when the center of enlargement is located inside the original object and the scale factor is between 0 and 1. The shape shrinks toward the internal center point.

Define 'Scale Factor' in the context of geometric transformations.

The scale factor is the constant multiplier applied to all linear dimensions of an object to create a similar image. It is calculated as the ratio of an image length to its corresponding object length.

What is the 'Center of Enlargement'?

The center of enlargement is the fixed point in a plane from which all points are expanded or contracted. It is the only point that does not move during the enlargement transformation.

State the formula for the area of an enlarged image given the original area $A$ and scale factor $k$.

The formula is $\text{Area}_{\text{image}} = k^2 \times \text{Area}_{\text{object}}$. This applies regardless of whether $k$ is positive, negative, or fractional.

How do you calculate the scale factor if you are given the coordinates of the object and image?

Divide the length of one side of the image by the length of the corresponding side on the object. Alternatively, divide the distance from the center to an image vertex by the distance from the center to the corresponding object vertex.

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Enlargements

Summary

Enlargement is a geometric transformation that changes the size of a shape while maintaining its proportions and internal angles. It is defined by two critical parameters: a center of enlargement, which acts as the fixed reference point for the transformation, and a scale factor, which determines the magnitude and direction of the size change.

1. Definition & Core Concepts

An enlargement is a transformation that produces an image that is the same shape as the original object but a different size, making the two shapes mathematically similar.

The Scale Factor ( $k$ ) is the ratio of the length of a side on the image to the corresponding length on the original object, expressed as $k = \frac{\text{image length}}{\text{object length}}$ .

The Center of Enlargement (CoE) is a fixed point from which the enlargement is measured; it determines the final position of the enlarged image relative to the original.

During an enlargement, all angles remain invariant, meaning they do not change, which is why the shape's overall appearance is preserved.

A diagram showing a center of enlargement with rays passing through a small triangle to a larger, similar triangle, illustrating a scale factor of 2.

2. The Role of the Scale Factor

3. Area and Volume Relationships

4. Step-by-Step Methodology

5. Key Distinctions

6. Exam Strategy & Tips

Enlargements

Summary

1. Definition & Core Concepts

An enlargement is a transformation that produces an image that is the same shape as the original object but a different size, making the two shapes mathematically similar.

The Center of Enlargement (CoE) is a fixed point from which the enlargement is measured; it determines the final position of the enlarged image relative to the original.

During an enlargement, all angles remain invariant, meaning they do not change, which is why the shape's overall appearance is preserved.

A diagram showing a center of enlargement with rays passing through a small triangle to a larger, similar triangle, illustrating a scale factor of 2.

2. The Role of the Scale Factor

If the scale factor $k > 1$ , the image is larger than the object and is located further away from the center of enlargement.

If the scale factor is a fraction between 0 and 1 ( $0 < k < 1$ ), the image is smaller than the object and is located closer to the center of enlargement.

A negative scale factor ( $k < 0$ ) causes the image to appear on the opposite side of the center of enlargement and results in the image being inverted (rotated $180^\circ$ ).

The magnitude of the scale factor determines the size change: for example, $k = -2$ means the image is twice as large, inverted, and on the opposite side of the center.

3. Area and Volume Relationships

While lengths are scaled by a factor of $k$ , the area of the shape increases by a factor of $k^2$ .

This relationship exists because area is a two-dimensional measure; if both base and height are multiplied by $k$ , the resulting area is $k \times k = k^2$ times the original.

Similarly, the volume of a three-dimensional object increases by a factor of $k^3$ when enlarged.

Understanding these ratios is vital for solving problems where only the change in capacity or surface area is provided.

4. Step-by-Step Methodology

Step 1: Identify the coordinates of the Center of Enlargement (CoE) and the Scale Factor ( $k$ ).
Step 2: For each vertex of the object, calculate the horizontal and vertical distance from the CoE.
Step 3: Multiply these distances by the scale factor $k$ to find the new relative distances.
Step 4: Starting from the CoE, apply these new distances to plot the corresponding vertices of the image.
Step 5: Connect the new vertices to complete the enlarged shape and verify that corresponding sides are parallel.

5. Key Distinctions

Feature	Positive Scale Factor ( $k > 0$ )	Negative Scale Factor ( $k < 0$ )
Orientation	Same as the original object	Inverted ( $180^\circ$ rotation)
Position	Same side of the Center	Opposite side of the Center
Distance	$k \times$ distance from Center	$

It is important to distinguish between congruence and similarity: enlargement creates similar shapes, whereas transformations like translation or reflection create congruent shapes.

6. Exam Strategy & Tips

Ray Method: Always draw 'projection lines' or rays from the center through the vertices of the object; the image vertices must lie on these same lines.
Check the Center: If asked to find the center of enlargement, draw lines connecting at least two pairs of corresponding vertices; the point where these lines intersect is the CoE.
Inverse Transformations: To return an enlarged image back to its original size, use an enlargement with the same center but a scale factor of $\frac{1}{k}$ .
Sanity Check: If $k$ is fractional, ensure your image is smaller; if $k$ is negative, ensure it is 'upside down' and on the other side of the center.