Transformations

6) Fractional enlargements and finding the centre of enlargement

Fractional enlargements

When a shape is enlarged by a scale factor between 0 and 1, the image is smaller than the original shape.

Triangle (ABC) is enlarged by 1/3 to triangle A'B'C'

The triangle ABC is enlarged by a scale factor of \frac{1}{3}. All the sides of triangle A’B’C’ are one third as long as the sides of the original triangle ABC.

Enlarge the triangle ABC by a scale factor of \frac{1}{2} about the centre of enlargement O.

Enlarge triangle (ABC) by 1/2

First, draw ray lines from O to each corner of the triangle.

Next, measure the distance from O to each corner of ABC. Divide the distance by two and plot the points A’ B’ and C’. Alternatively these distances can be shown as vectors. OA = \left(\begin{array}{c}2\ 2\end{array}\right) so under a scale factor of \frac{1}{2}, OA’ = \left(\begin{array}{c}1\ 1\end{array}\right).

Finally, join up the points A’ B’ C’.

Triangle A'B'C' produced after enlarging triangle by 1/2

Finding the centre of enlargement

To find the centre of enlargement, draw ray lines from the corners of the image through the corners of the original shape.

Describe the transformation of the triangle RST.

Triangle (STR) and triangle (S'T'R)

Draw ray lines from the corners of triangle RST through the corners of R’S’T’ until they cross. This is the centre of enlargement.

Triangles (STR) and (S'T'R') with lines of enlargement

The triangle has been enlarged by a scale factor of \frac{1}{2} about the centre of enlargement (0,2).


Negative enlargements

An enlargement with a negative scale factor produces an image on the other side of the centre of enlargement. The image appears upside down.

Rectangle (ABCD) enlarged by -1/2 to produce rectangle (A'B'C'D')

The rectangle ABCD has been enlarged by a scale factor of \frac{-1}{2}.

The lengths in rectangle A’B’C’D’ are \frac{1}{2} times as long as rectangle ABCD. The distance from O to A’B’C’D’ is half the distance from O to ABCD.

Question 1
Shape (EFGH)

Enlarge the quadrilateral EFGH by a scale factor of -½ about the centre of enlargement O


Dotted lines drawn from in each point, through the origin

First, draw ray lines from each corner of the quadrilateral to O and extend them


Image (E'F'G'H') drawn at the new points

Next, measure the distance from O to each corner of EFGH. Divide the distance by two and plot the points E’ F’ G’ and H’ on the other side of the centre of enlargement. Finally, join up the points E’ F’ G’ and H’

An enlargement by a scale factor of -1 is the same as a rotation of 180°.

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