Recognise an enlargement and similar shapes

Key Rules:

• 1. Two shapes are similar when one is an enlargement of the other: matching angles are equal and matching sides are all in the same ratio.
• 2. Find the scale factor by dividing a length on the new shape by the matching length on the original.

Remember:

• •Enlargement: scale factor and centre. Rotation: angle, direction and centre. Similar means equal angles and sides in one shared ratio.

Don’t Forget...

• •Describing an enlargement with a scale factor but no centre. Scale factor 3 from centre (0, 0) and scale factor 3 from centre (2, 1) put the image in different places, so the description does not fix a single transformation until the centre is named.
• •Testing similarity by adding to sides instead of multiplying. A 2 by 4 rectangle is similar to a 4 by 8 rectangle because both sides double; a 4 by 6 rectangle has both sides increased by 2, but 42 and 64 are different ratios, so it is not similar.
• •Giving a rotation's angle without its direction or centre. A 90 degree turn clockwise and a 90 degree turn anticlockwise about the same centre give different images, and moving the centre moves the image, so angle, direction and centre are all needed (only a 180 degree turn needs no direction).

Worked Example:

Triangle A has corners at (1, 1), (3, 1) and (1, 2). Triangle B has corners at (3, 3), (9, 3) and (3, 6). Describe fully the single transformation that takes A to B.

Compare matching sides: the base of A from (1, 1) to (3, 1) has length 2, and the base of B from (3, 3) to (9, 3) has length 6, so the scale factor is 62 = 3.

Check a second pair: the left side of A has length 1 and the matching side of B has length 3, so the factor 3 holds for every side.

Find the centre: the straight line through the matching corners (1, 1) and (3, 3), and the line through (3, 1) and (9, 3), both pass through (0, 0).

Name all three parts of the description: the word enlargement, the scale factor and the centre.

Answer: An enlargement with scale factor 3, centre (0, 0)

Why It Works:

An enlargement multiplies every point's distance from the centre by the scale factor, so all lengths grow in the same ratio while every angle is untouched, which is exactly what similar means. A rotation carries every point around a circle centred at the centre of rotation through the same angle, so no length or angle inside the shape can change.

Another Example:

Triangle P has corners at (1, 1), (4, 1) and (1, 3). It is transformed onto triangle Q with corners at (-1, 1), (-1, 4) and (-3, 1). Describe the transformation fully.

P and Q are the same size, so this is not an enlargement; the shape has turned, so test a rotation.

Follow one corner: (4, 1) has moved to (-1, 4), which is a quarter turn anticlockwise about the origin, because that turn sends (x, y) to (-y, x).

Check the other corners: (1, 1) lands on (-1, 1) and (1, 3) lands on (-3, 1), matching Q exactly.

State all four parts: rotation, 90 degrees, anticlockwise, centre (0, 0).

Answer: A rotation of 90 degrees anticlockwise about (0, 0)