Enlargement, similarity, and combined transformations
Animation: triangle enlarged by scale factor 2 from the origin. Lines from O pass through each vertex to show how enlargement works.
What is Enlargement?
An enlargement changes the size of a shape but keeps its angles the same and all sides in the same ratio. The enlarged shape is called the image. The original is called the object.
Every enlargement needs two things:
Scale factor (SF) — how many times bigger (or smaller)
Centre of enlargement — the fixed point from which distances are measured
Image distance from centre = SF × Object distance from centre
🔍
SF > 1 — Gets Bigger
SF = 2 means every length doubles. SF = 3 means every length triples.
🔬
0 < SF < 1 — Gets Smaller
SF = ½ means every length halves. The image is inside the object.
📐
Finding the Centre
Draw lines through matching vertices of object and image — they all meet at the centre.
⚖️
Finding the SF
SF = image side length ÷ object side length. Or: image distance from centre ÷ object distance from centre.
Similar Shapes
Two shapes are similar if one is an enlargement of the other. Similar shapes have:
All corresponding angles equal
All corresponding sides in the same ratio
Missing side = Known side × Scale Factor | SF = image side ÷ object side
To find a missing length: identify the ratio (scale factor) from a pair of known corresponding sides, then multiply.
Combined Transformations
Two transformations applied in sequence. Order matters! Always apply the first transformation, then use that result as the new object for the second transformation.
Common combinations: reflect then enlarge, translate then rotate, enlarge then reflect.
Apply Transformation 1 → Get Image 1 → Apply Transformation 2 → Get Final Image
Key Vocabulary
Term
Meaning
Object
The original shape before transformation
Image
The shape after transformation
Scale factor
The multiplier for all lengths
Centre of enlargement
The fixed point from which enlargement is measured
Similar
Same shape, different size (all angles equal, sides in ratio)
Congruent
Same shape AND same size (scale factor = 1)
Worked Examples
Example 1 — Enlarge by Scale Factor 2 from the Origin
Triangle with vertices A(1,1), B(3,1), C(1,3). Enlarge by SF 2 from centre O(0,0).
Rule: Multiply each coordinate by the scale factor.
A(1,1) → A'(2,2) [1×2, 1×2]
B(3,1) → B'(6,2) [3×2, 1×2]
C(1,3) → C'(2,6) [1×2, 3×2]
Check: Every side of A'B'C' is twice as long. Angles unchanged.
Example 2 — Enlarge by Scale Factor ½ from Centre (4,4)
Square with vertices P(2,2), Q(6,2), R(6,6), S(2,6). Enlarge by SF ½ from centre C(4,4).
Rule: Find the vector from centre to each vertex, then multiply by SF.
P(2,2): vector from C = (2−4, 2−4) = (−2,−2). Multiply by ½ = (−1,−1). New point: (4−1, 4−1) = P'(3,3)
Find SF: Compare corresponding side lengths. AB = 2 units; A'B' = 4 units. SF = 4 ÷ 2 = 2
Find centre: Draw lines through A and A', B and B', C and C'. They all pass through (1,1)... extend the lines.
Line through A(1,1) and A'(−1,−1): this is the line y = x. Line through B(3,1) and B'(−5,−1): slope = (−1−1)/(−5−3) = ¼, passes through... checking: all lines meet at O(0,0).
Answer: Enlargement, SF = 2, centre (0,0). Note: negative image means centre is between object and image (or object on far side).
Example 4 — Missing Length in Similar Triangles
Triangle ABC is similar to triangle DEF. AB = 6 cm, BC = 8 cm, AC = 10 cm. DE = 9 cm. Find EF.
Step 1: Find the scale factor. AB corresponds to DE (both are the "short" sides). SF = DE ÷ AB = 9 ÷ 6 = 1.5
Step 2: EF corresponds to BC. EF = BC × SF = 8 × 1.5 = 12 cm
Step 3: DF corresponds to AC. DF = 10 × 1.5 = 15 cm
Check: All sides in ratio 6:9 = 8:12 = 10:15 = 2:3. ✓
Enlargement Explorer
Adjust the scale factor and centre to see how the shape transforms live.
SF = 1.5
Blue = object (fixed triangle), Pink = enlarged image, Orange dot = centre of enlargement. Orange lines show how each vertex is projected.
Try these:
Set SF = 2, Centre (0,0) — classic enlargement from origin
Set SF = 0.5, Centre (0,0) — image is half the size
Set SF = 2, Centre (2,1) — enlargement from a different point
Set SF = 1.0 — image exactly overlaps object (no change)
Exercise 1 — Scale Factor Finder
Each pair of shapes is an enlargement. Find the scale factor. Type your answer (e.g. 2 or 0.5 or 1/3).
Exercise 2 — Enlargement on a Grid
Each question shows a shape (blue). Type the coordinates of each vertex of the enlarged image. Use the diagram as reference.
Exercise 3 — Similar Shapes: Missing Lengths
Each pair of shapes is similar. Find the missing length. Round to 1 decimal place if needed.
Exercise 4 — Centre of Enlargement Finder
Click on the grid where you think the centre of enlargement is. You're correct if you click within 1 grid square.
Exercise 5 — Describe the Transformation
Each diagram shows an object and its image. Describe the enlargement fully: type, scale factor, and centre of enlargement.
Exercise 6 — Combined Transformations
Apply two transformations in sequence. Type the final vertex coordinates after both transformations.
Practice Questions
20 questions — reveal each answer when you're ready.