5.2 Enlargement
Enlargement is a geometric transformation where all points of an object move from a fixed point by a constant ratio known as the scale factor. The image produced is geometrically similar to the original object, and its area is related to the square of the scale factor.
Learning Objectives
- Explain the meaning of similarity of geometric objects.
- Make a connection between similarity and enlargement, hence describe enlargement using the centre of enlargement and scale factor.
- Determine the image and object of an enlargement using projection lines and coordinate distances.
- Make and verify conjectures on the relation between the area of the image and the area of the object of an enlargement.
- Solve problems involving enlargement, scale factors, side lengths, and areas.
Summary of Scale Factor Properties in Enlargement
See how the magnitude and sign of scale factor \(k\) affect the size, orientation and position of the image relative to the centre of enlargement \(P\).
Larger image, same side
The image is larger than the object and lies on the same side of centre \(P\) as the object.
Smaller image, same side
The image is smaller than the object and lies on the same side of centre \(P\) as the object.
Smaller image, inverted
The image is smaller than the object, lies on the opposite side of centre \(P\), and is inverted.
Larger image, inverted
The image is larger than the object, lies on the opposite side of centre \(P\), and is inverted.
Brief Explanation
What is Enlargement?
Enlargement is a transformation in which all the points of an object move from a fixed point with a constant ratio. The fixed point is called the centre of enlargement, \(P\), and the constant ratio is the scale factor, \(k\).
Similarity and Enlargement
Two geometric objects are similar when all corresponding angles are equal and all ratios of corresponding sides are equal. An object and its image under an enlargement are always similar.
\[k = \frac{\text{distance of point of image from } P}{\text{distance of point of object from } P} = \frac{PA'}{PA}\]
\[k = \frac{\text{length of side of image}}{\text{length of side of object}}\]
| Scale factor, \(k\) | Size of image | Position relative to centre \(P\) | Orientation |
|---|---|---|---|
| \(k > 1\) | Larger than object | Same side | Same orientation |
| \(k = 1\) | Equal in size | Same side | Same orientation |
| \(0 < k < 1\) | Smaller than object | Same side | Same orientation |
| \(-1 < k < 0\) | Smaller than object | Opposite side | Inverted |
| \(k = -1\) | Equal in size | Opposite side | Inverted |
| \(k < -1\) | Larger than object | Opposite side | Inverted |
\[\frac{\text{Area of image}}{\text{Area of object}} = k^2\]
\[\text{Area of image} = k^2 \times \text{Area of object}\]
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
A triangle with an area of \(18\text{ cm}^2\) is enlarged with a scale factor of \(3\). What is the area of the image?
The length of a side of an object is \(6\text{ cm}\) and the corresponding side of its image is \(15\text{ cm}\). Calculate the scale factor of the enlargement.
In an enlargement with centre \(P\), the distance of object vertex \(A\) from \(P\) is \(4\text{ units}\). The corresponding image vertex \(A'\) lies on the opposite side of \(P\) at a distance of \(8\text{ units}\). What is the scale factor \(k\)?
Common Mistakes
- Using \(k\) instead of \(k^2\) when calculating the area of an enlarged figure.
- Inverting the scale factor formula by calculating \(\frac{\text{object}}{\text{image}}\) instead of \(\frac{\text{image}}{\text{object}}\).
- Forgetting to include the negative sign for \(k\) when the image is inverted and on the opposite side of the centre of enlargement.
- Describing an enlargement incompletely by stating only the scale factor while omitting the centre of enlargement coordinates.
Misconceptions
An enlargement always makes a shape larger.
In mathematics, “enlargement” includes reductions where \(0 < |k| < 1\), which produce an image smaller than the object.
Negative scale factor means the area of the image is negative.
Area is always positive because the area scale factor is \(k^2\), and squaring any real negative number produces a positive number.
Any two rectangles with proportional perimeters are similar.
For quadrilaterals, both corresponding angles must be equal AND all corresponding side ratios must be equal.
Summary
- Similarity: equal angles and proportional sides, \(\frac{A'B'}{AB} = k\).
- \(k = \frac{\text{Image distance from } P}{\text{Object distance from } P} = \frac{\text{Length of image side}}{\text{Length of object side}}\).
- \(\text{Area of image} = k^2 \times \text{Area of object}\).
- Description format: Enlargement at centre \(P(x, y)\) with scale factor \(k\).
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 An object with an area of \(25\text{ cm}^2\) undergoes an enlargement with a scale factor of \(k = -3\). What is the area of the image?
2 Which of the following scale factors results in an image that is smaller than the object and located on the opposite side of the centre of enlargement?
B. Fill in the Blanks
3 Under an enlargement, the ratio of the area of the image to the area of the object is equal to .
Answer for question 34 When describing an enlargement, one must specify the scale factor and the of enlargement.
Answer for question 4C. Drag and Drop: Match Scale Factor to Image Characteristics
Match each scale factor range to the correct size and position of the resulting image relative to the centre of enlargement \(P\). On a phone, tap the answer choice first, then tap the matching box.
