Properties and constructions: including fractional scale factors

The idea

A fractional scale factor enlarges or reduces a shape by multiplying every distance from the centre of enlargement by a fraction. A proper fraction (between 0 and 1) produces a smaller image, while an improper fraction (greater than 1) produces a larger image. For example, enlarging a shape by scale factor 1/2 from centre (0, 0) maps each vertex (x, y) to (x/2, y/2), whereas a scale factor of 3/2 maps it to (1.5x, 1.5y). The image is similar to the object: corresponding angles are equal and corresponding sides are in the same ratio. Every image vertex lies on the ray from the centre through its object vertex. Fractional factors may be given as decimals or percentages (e.g. 50% becomes 0.5), so convert carefully before multiplying.

What earns the marks

• A fractional scale factor between 0 and 1 produces a smaller image, while a fractional scale factor greater than 1 (an improper fraction) produces a larger image.
• Every vertex of the image lies on the ray from the centre of enlargement through the corresponding object vertex.
• For a positive fractional scale factor, the image lies on the same side of the centre as the object.
• Corresponding angles are unchanged and corresponding side lengths are multiplied by the scale factor.
• Coordinates transform as (x, y) → (kx, ky) when the centre is the origin, where k is the fractional scale factor.
• A scale factor given as a decimal or percentage must be converted to a fraction or decimal before use.

Common mistakes and how to put them right

• Dividing by the scale factor instead of multiplying.

  Put it right: For scale factor 1/2, multiply each distance from the centre by 1/2, not by 2.

• Assuming all fractional scale factors make a shape smaller.

  Put it right: Improper fractions like 3/2 will enlarge the shape.

• Changing the angles of the shape.

  Put it right: Enlargement preserves angles, so only lengths change.

• Treating a percentage such as 50% as 50 rather than 0.5 before multiplying.

Exam tips

• Draw construction rays from the centre through each object vertex before marking image vertices.
• Measure or calculate each image distance as the object distance multiplied by the fractional scale factor.
• Verify one pair of corresponding sides has the correct ratio and that all angles match.