Enlargement

Enlargement is a transformation in which a figure is made larger (magnified) or made smaller (diminished). A photograph may be enlarged or diminished to suit a certain purpose. Figures can be drawn to scale where actual figures are diminished or enlarged. Enlarged shapes are geometrically similar and have corresponding angles equal. The number that magnifies or diminishes a figure is called the enlargement factor and is usually denoted by $k$. If $k$ is less than 1 the figure is diminished and if it is greater than 1 the figure is enlarged $k$ times.

In the case of closed figures, if the lengths are enlarged by a factor of $k$ then the area is enlarged by $k^2$.

Similarity can be used in enlarging or diminishing geometrical figures. For example in maps, a large area of land is represented by a small area on paper by a scale. Scale is a ratio between the measurement of a drawing to the actual measurement. It is normally stated in the form of 1 : n, for example, if a scale of a map is 1:20000, then 1 unit on the map represents 20000 units on the ground.

In the figure, triangle ABC is a scale measurement of triangle PQR, where the scale is 1:2.

Scale is measurement of drawing: Actual Measurement, that is,

$Scale = \frac{\text{Measurement of drawing}}{\text{Actual measurement}}$

A building 250 metres high is represented by a line segment of length 5 cm. Find the scale of the drawing.

A triangular plot of land has sides 152 metres and 208 metres meeting at an angle of 50°. Find by scale drawing the distance from the middle point of the longest side to the opposite corner

Consider two similar rectangles shown in the figure below with a scale factor $k$.

If $AB = a$ and $AD = b$, then $PO = ak$ and $PS = bk$

Area of $ABCD = a \times b = ab$

Area of $PORS = ak \times bk = abk^2$

$\frac{\text{Area of PORS}}{\text{Area of ABCD}} = \frac{abk^2}{ab} = k^2$

Therefore, if two polygons have a scale factor $k$ then the ratio of their areas is $k^2$. This is also called the scale factor for the area.

If we 'enlarge' a shape by a scale factor that is between -1 and 1, the image will be smaller than the object

Example 2

Enlarge triangle ABC with a scale factor $\frac{1}{2}$, centred about the origin.

Solution

The scale factor is $\frac{1}{2}$, so:

$OA' = \frac{1}{2}OA$

$OB' = \frac{1}{2}OB$

$OC' = \frac{1}{2}OC$

Since the centre is the origin, we can in this case multiply each coordinate by $\frac{1}{2}$ to get the answers.

$A = (2, 2)$, so $A'$ will be $(1, 1)$.

$B = (2, 6)$, so $B'$ will be $(1, 3)$.

$C = (4, 2)$, so $C'$ will be $(2, 1)$.

If two polygons are similar and the ratio of their corresponding sides of two similar polygons is 5:3, then the scale of enlargement is $\frac{5}{3}$