Enlargement and Reduction of Geometric Shapes

Learn what a point is, see how it’s defined, then watch the equation of the circle get derived — one step at a time.

Introduction to Enlargement and Reduction

An enlargement or a reduction (together called a dilation) is a transformation that changes the size of a geometric figure without changing its shape. It is defined by a fixed point called the centre of enlargement and a number $k$ called the scale factor. Every point of the image lies on the line joining the centre to the corresponding point of the object, at a distance $k$ times as far from the centre.

In this topic, we will learn the following standard cases of enlargement and reduction in the Cartesian plane:

Enlargement ($k > 1$) – the image is larger than the object and lies farther from the centre than the object does.
Reduction ($0 < k < 1$) – the image is smaller than the object and lies closer to the centre than the object does.
Scale factor $k = 1$ – the image is identical to the object; the figure does not change.
Negative scale factor ($k < 0$) – the image lies on the opposite side of the centre and appears inverted. It is an enlargement if $|k| > 1$ and a reduction if $|k| < 1$.
Centre at the origin – transforms a point $(x, y)$ to $(kx, ky)$ by multiplying both coordinates by $k$.
Centre at the point $(a, b)$ – transforms a point $(x, y)$ to $\big(a + k(x - a),\ b + k(y - b)\big)$.

Scale Factor

The scale factor is the ratio of a length in the image to the corresponding length in the object:

$$k = \frac{\text{length of a side in the image}}{\text{length of the corresponding side in the object}} = \frac{CA'}{CA}$$

where $C$ is the centre of enlargement, $A$ is a point of the object and $A'$ is its image.

Properties of Enlargement and Reduction

The object and its image are similar: they have the same shape, but they are congruent only when $k = 1$ or $k = -1$.
Corresponding angles of the object and the image are equal.
Corresponding sides are proportional, so every length in the image is $|k|$ times the corresponding length in the object.
The centre of enlargement, a point of the object and its image are collinear (they lie on one straight line).
Corresponding sides of the object and the image are parallel.
The perimeter of the image is $|k|$ times the perimeter of the object.
The area of the image is $k^2$ times the area of the object.
For $k > 0$ the object and its image are on the same side of the centre and have the same orientation; for $k < 0$ they are on opposite sides and the image is inverted.
The centre of enlargement remains invariant (unchanged).

Enlargement when the centre of enlargement is at the origin O(0, 0)

Step 1 of 6

Enlargement with the centre at the origin O(0, 0)

Definition:

Enlargement with the centre of enlargement at the origin O(0, 0) and scale factor k changes the size of a figure without changing its shape. Each point P(x, y) moves along the line OP so that its distance from the origin becomes k times the original distance. The new coordinates are obtained by multiplying both the x-coordinate and the y-coordinate by k.

1. Problem:

In the given graph, the triangle ABC with vertices A(1, 1), B(5, 2) and C(2, 4) is enlarged with the centre of enlargement at the origin O and scale factor k = 2. The enlarged triangle A'B'C' is shown in the graph. What are the coordinates of A', B' and C'? Compare and draw a conclusion.

2. Procedure:

Enlarge the points A(1, 1), B(5, 2) and C(2, 4) with the centre at the origin and scale factor 2, write the coordinates of the enlarged points A', B' and C' and discuss on your conclusion.

3. Points (A, B, C and A', B', C'):

A(1,1)E_o: [O, k = 2]A'(kx, ky) = A'(2, 2)

B(5,2)E_o: [O, k = 2]B'(kx, ky) = B'(10, 4)

C(2,4)E_o: [O, k = 2]C'(kx, ky) = C'(4, 8)

4. Formula:

Thus, P(x, y) — Enlarge: [O, k] → P'(kx, ky)

5. Conclusion:

Under an enlargement with centre at the origin and scale factor k, every point P(x, y) moves to P'(kx, ky). So A, B and C move to A'(2, 2), B'(10, 4) and C'(4, 8). Each image point lies on the line joining the origin to the original point, at k = 2 times the distance from the origin. The sides of the triangle become 2 times as long and stay parallel to the original sides, so the shape stays the same while the size changes.

Enlargement when the centre of enlargement is at any point Q(a, b)

Step 1 of 6

Enlargement with the centre at any point Q(a, b)

Definition:

Enlargement with the centre of enlargement at any point Q(a, b) and scale factor k changes the size of a figure without changing its shape. Each point P(x, y) moves along the line QP so that its distance from the centre Q becomes k times the original distance. The new coordinates are obtained by multiplying the horizontal and vertical distances from Q by k: P'(a + k(x − a), b + k(y − b)).

1. Problem:

In the given graph, the triangle ABC with vertices A(1, 2), B(–1, 2) and C(–1, –1) is enlarged with the centre of enlargement at Q(–2, 1) and scale factor k = 3. The enlarged triangle A'B'C' is shown in the graph. What are the coordinates of A', B' and C'? Compare and draw a conclusion.

2. Procedure:

Enlarge the points A(1, 2), B(–1, 2) and C(–1, –1) with the centre at Q(–2, 1) and scale factor 3, write the coordinates of the enlarged points A', B' and C' and discuss on your conclusion.

3. Points (A, B, C and A', B', C'):

A(1, 2)E_Q: [Q(–2, 1), k = 3]A'(7, 4)

A' = (–2 + 3{1 – (–2)}, 1 + 3(2 – 1)) = (7, 4)

B(–1, 2)E_Q: [Q(–2, 1), k = 3]B'(1, 4)

B' = (–2 + 3{–1 – (–2)}, 1 + 3(2 – 1)) = (1, 4)

C(–1, –1)E_Q: [Q(–2, 1), k = 3]C'(1, –5)

C' = (–2 + 3{–1 – (–2)}, 1 + 3(–1 – 1)) = (1, –5)

4. Formula:

Thus, P(x, y) — Enlarge: [Q(a, b), k] → P'(a + k(x − a), b + k(y − b))

5. Conclusion:

Under an enlargement with centre Q(a, b) and scale factor k, every point P(x, y) moves to P'(a + k(x − a), b + k(y − b)). So A, B and C move to A'(7, 4), B'(1, 4) and C'(1, –5). Each image point lies on the line joining the centre Q(–2, 1) to the original point, at k = 3 times the distance from Q. The centre Q itself does not move. The sides of the triangle become 3 times as long and stay parallel to the original sides, so the shape stays the same while the size changes.

Example 1 of 2

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