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.
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 \(P\).
The image is larger than the object and lies on the same side of centre \(P\) as the object.
The image is smaller than the object and lies on the same side of centre \(P\) as the object.
The image is smaller than the object, lies on the opposite side of centre \(P\), and is inverted.
The image is larger than the object, lies on the opposite side of centre \(P\), and is inverted.
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, \(k\).
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.
\[k = \frac{\text{distance of point of image from } P}{\text{distance of point of object from } P} = \frac{PA'}{PA}\]
\[k = \frac{\text{length of side of image}}{\text{length of side of object}}\]
\[\frac{\text{Area of image}}{\text{Area of object}} = k^2\]
\[\text{Area of image} = k^2 \times \text{Area of object}\]
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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 \(k\)?
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.
Complete all parts first, then press the check answer button.
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?
3 Under an enlargement, the ratio of the area of the image to the area of the object is equal to .
4 When describing an enlargement, one must specify the scale factor and the of enlargement.
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.
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