Enlargement with a Center

Rays from the center

An enlargement changes the size of a shape by a scale factor, as in Scale Factor: every length is multiplied by the same number. What decides where the image goes is a fixed point called the center of enlargement.

From the center, draw a ray, a half-line that starts at the center, through each vertex of the shape. The image of each vertex lies on its own ray. With a scale factor bigger than 1, it sits farther out along the ray than the vertex.

k times as far from the center

With scale factor k, each image vertex is k times as far from the center as the vertex it came from, along the same ray. With the origin as the center and k = 2, the vertex (1, 1) goes to (2, 2), (2, 1) goes to (4, 2), and (2, 2) goes to (4, 4).

Check one distance. (2, 1) is $\sqrt{2^2+1^2}=\sqrt{5}$√(2² + 1²) = √5 from the origin, and its image (4, 2) is $\sqrt{4^2+2^2}=\sqrt{20}=2\sqrt{5}$√(4² + 2²) = √20 = 2√5, twice as far. Because every distance from the center is doubled, every side is doubled too: the side from (1, 1) to (2, 1) is 1 long, and its image, from (2, 2) to (4, 2), is 2 long. The angles do not change, so the image is similar to the object.

An enlargement with scale factor 2 and the origin, the gold dot, as its center. The faint dashed rays run from the center through each corner of A to the matching corner of A', which is twice as far out.

A center that is not the origin

When the center is somewhere else, work with the step from the center to each vertex, written as a column vector. Multiply the step by k, then take the new step from the center.

Take the center (1, 2), the scale factor 3, and the vertex (3, 3). The step from the center to the vertex is [3 − 1, 3 − 2] = [2, 1]. Three times that is [6, 3], and from the center it reaches (1 + 6, 2 + 3) = (7, 5). The other vertices of the triangle go the same way: (2, 3) goes to (4, 5), and (2, 4) goes to (4, 8).

Multiplying the coordinates themselves, (3, 3) × 3 = (9, 9), is correct only when the center is the origin, because only then is the step from the center the same as the coordinates.

An enlargement with scale factor 3 and center (1, 2), the gold dot. The triangle with corners (2, 3), (3, 3) and (2, 4) goes to the triangle with corners (4, 5), (7, 5) and (4, 8).

A scale factor between 0 and 1

With $k=\frac{1}{2}$k = 1/2, each vertex lands half as far from the center, on the same ray. The image is smaller than the object and sits between the object and the center. With the origin as the center, (2, 2) goes to (1, 1), (4, 2) goes to (2, 1), and (4, 4) goes to (2, 2), and every side is half as long.

It is still called an enlargement. The word names the transformation, multiplying distances from a center, whatever the size of k.

An enlargement with scale factor $\frac{1}{2}$1/2 from the origin. Each corner of the image is halfway along the ray from the center to the matching corner of the object.

A negative scale factor

A negative scale factor measures the distance out on the other side of the center. Multiply the step from the center by k: with k = −1, the step [1, 1] becomes [−1, −1], which points the opposite way. So each image vertex is as far from the center as before, but on the far side, and the line from each vertex to its image runs through the center.

With the origin as the center and k = −1, (1, 1) goes to (−1, −1), (3, 1) goes to (−3, −1), and (3, 2) goes to (−3, −2). The image is upside down: k = −1 has the same effect as a half turn about the center. With k = −2, the image is upside down and twice as big: (1, 1) goes to (−2, −2) and (3, 1) goes to (−6, −2).

An enlargement with scale factor −1 from the origin. Each dashed line runs from a corner through the center to the matching corner on the far side, and the image is upside down.

Finding the center and the scale factor

Every ray passes through the center. So to find the center from a shape and its image, join each vertex to its image with a straight line and extend the lines: they all meet at the center.

The triangle with corners A(2, 3), B(3, 3) and C(2, 1) has the image A'(3, 2), B'(5, 2) and C'(3, −2). The lines through A and A', through B and B', and through C and C' all pass through (1, 4), which is the center.

The scale factor is a length of the image divided by the matching length of the object. AB is 3 − 2 = 1 long and A'B' is 5 − 3 = 2 long, so $k=2/1=2$k = 2 / 1 = 2. Check from the center: A is the step [1, −1] from (1, 4), and A' is the step [2, −2], twice as far along the same ray.

The triangle with corners (2, 3), (3, 3) and (2, 1), and its image with corners (3, 2), (5, 2) and (3, −2). The dashed lines joining matching corners all meet at the gold dot, (1, 4), the center.

Lengths, angles and area

An enlargement with scale factor k multiplies every length by k, or by its size when k is negative, and leaves every angle unchanged. Area is multiplied by $k^2$k², because both the width and the height are multiplied by k.

In the triangle ABC, the sides AB and AC are 1 and 2 long and meet at a right angle, so its area is $\frac{\left(1\times 2\right)}{2}=1$(1 × 2)/2 = 1. In its image the matching sides are 2 and 4, and the area is $\frac{\left(2\times 4\right)}{2}=4$(2 × 4)/2 = 4, which is $2^2=4$2² = 4 times as much.

The usual mistakes

Adding the scale factor. A vertex 2 units from the center, with scale factor 3, lands 2 × 3 = 6 units out, not 2 + 3 = 5.

Squaring the scale factor for a length. Lengths are multiplied by k; only areas are multiplied by $k^2$k². With scale factor 3, 2 units becomes 6, not 18.

Reading a fraction as a bigger shape. $k=\frac{1}{2}$k = 1/2 halves every distance from the center; it does not double anything, and it does not slide the shape.

Putting the image of a negative scale factor on the same side. A negative k sends each vertex through the center to the far side.

Measuring from the wrong point. Every distance in an enlargement is measured from the center, and when the center is not the origin, the coordinates themselves are not multiplied.