top of page

19.2 More Similarity

GCSE Maths Foundation · Edexcel · Chapter 19: Congruence, Similarity and Vectors

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: similar triangles with parallel lines, and enlargements from a centre. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Spot similar triangles made by parallel lines
  • Find missing lengths in triangles inside triangles and in crossing ("bow-tie") diagrams
  • Enlarge a shape from a centre of enlargement, including by a fractional scale factor
  • Find the scale factor and centre of an enlargement and describe it fully

Key words

  • Parallel: Lines that are always the same distance apart. They are marked with matching arrows.
  • Centre of enlargement: The fixed point that the enlargement is measured from.
  • Fractional scale factor: A scale factor between 0 and 1. The image is smaller than the original, but it is still called an enlargement.
  • Vertically opposite: Angles opposite each other where two straight lines cross. They are equal.

A triangle inside a triangle

If DE is parallel to BC, then triangle ADE and triangle ABC are similar. They share angle A, and the angles at D and B are corresponding angles on parallel lines, so they are equal.

Worked example: find x

Draw the two triangles separately. Small triangle ADE, big triangle ABC.
AB = AD + DB = 4 + 6 = 10 cm. Scale factor = AB ÷ AD = 10 ÷ 4 = 2.5
x = BC = DE × 2.5 = 6 × 2.5 = 15 cm

Answer: x = 15 cm

Key fact: Use the whole side AB = 10 cm, not the part DB = 6 cm. The big triangle's side is the full length.

Crossing lines (the "bow-tie")

When AE and BD cross at C and AB is parallel to DE, triangles ABC and EDC are similar. The angles at C are vertically opposite, and angles BAC and DEC are alternate angles.

Worked example: find y

AC matches EC. Scale factor = 9 ÷ 6 = 1.5
DE matches AB, so y = 8 × 1.5 = 12 cm

Answer: y = 12 cm

Enlarging from a centre

To enlarge by scale factor k from centre O, find how far each corner is from O (count across and up). Multiply both distances by k to find the new corner.

Worked example: one corner at a time

O = (1, 1). Corner (4, 2) is 3 right and 1 up from O.
Scale factor 2: 6 right and 2 up from O, which is (1 + 6, 1 + 2) = (7, 3).
Repeat for (2, 2) → (3, 3) and (2, 3) → (3, 5), then join the corners.

Answer: Image corners (3, 3), (7, 3), (3, 5)

Fractional scale factors and describing an enlargement

A scale factor of 1/2 halves every length and every distance from the centre. To describe an enlargement fully, give three things: the word enlargement, the scale factor and the centre.

Worked example: describe the enlargement P → Q

Width of P = 6 squares, width of Q = 3 squares. Scale factor = 3 ÷ 6 = 1/2
Join matching corners and extend the lines. They meet at O = (0, 7).

Answer: Enlargement, scale factor 1/2, centre (0, 7)

Common mistakes

  • Using the part DB instead of the whole side AB when the small triangle sits inside the big one.
  • Matching the wrong sides in a bow-tie. Use the parallel lines: the side opposite angle C in one triangle matches the side opposite C in the other.
  • Thinking a scale factor of 1/2 makes the shape bigger. Any scale factor between 0 and 1 makes it smaller.
  • Leaving out the centre or the scale factor when asked to "describe fully".

Revision questions

Try these, then check your answers below each one.

Question 1. In triangle ABC, D is on AB and E is on AC, with DE parallel to BC. AD = 6 cm, DB = 3 cm and DE = 8 cm. Work out BC.

Answer: 12 cm

Working: AB = 6 + 3 = 9. Scale factor = 9 ÷ 6 = 1.5. BC = 8 × 1.5 = 12 cm.

Question 2. The point (3, 4) is enlarged by scale factor 2 with centre (1, 1). Where is its image? (Choose from: (5, 7), (6, 8), (4, 5), (7, 10).)

Answer: (5, 7)

Working: (3, 4) is 2 right and 3 up from (1, 1). Doubled: 4 right, 6 up, so the image is (1 + 4, 1 + 6) = (5, 7).

Question 3. A shape is enlarged by scale factor 1/3. One side of the shape is 18 cm. How long is this side in the image?

Answer: 6 cm

Working: 18 × 1/3 = 6 cm.

Related lessons

Continue in the full course

This lesson is section 19.2 (EGCMF1902) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

bottom of page