Year 9 Maths The Thai Guy

Lesson 13.1 · 25 min

Enlargement

Enlarging a shape from a centre by a scale factor, finding the centre and scale factor, and what happens to perimeter and area.

In this lesson you will learn to…

  • Enlarge a shape using a centre and a scale factor, including fractional scale factors
  • Find the scale factor and the centre of an enlargement
  • Know that lengths and perimeters multiply by
  • Know that areas multiply by

Why this matters

Enlargements are how photographs are resized, maps are drawn to scale and models are built. Understanding the effect on area matters too: a poster twice as wide needs four times as much paper.

Virenda’s mistake

Virenda’s trapezium has perimeter 16 cm and area 16 cm². He enlarges it until the perimeter is 96 cm and thinks the area must be 96 cm² too. The scale factor is 96÷16=696 \div 16 = 6, so the area is 16×62=57616 \times 6^2 = 576 cm², not 96.

Finding a scale factor from areas

A hexagon of area 21 cm² is enlarged to 525 cm². The area has multiplied by 525÷21=25525 \div 21 = 25, so k2=25k^2 = 25 and k=5k = 5.

Congruent or similar? An enlargement changes the size but not the shape, so the image is similar to the object. Reflections, rotations and translations give congruent images.

Key rules and formulas

  • Two things define an enlargement

    The centre of enlargement and the scale factor. Every point of the image is times as far from the centre as the matching point of the object, in the same direction.

  • Counting squares

    From O to A is 2 right and 2 up. With scale factor 2, from O to A′ is 4 right and 4 up.

  • Scale factors less than 1

    A scale factor between 0 and 1 makes the shape smaller and moves it towards the centre. It is still called an enlargement.

  • Perimeter and area

    Every length is multiplied by , so the perimeter is too. Area uses two lengths multiplied together, so it grows by .
Try it

Enlargement explorer

Scale factor
2
Centre of enlargement
(1, 1)
Lengths
every length is 2 times as long
Area
the area is 4 times as big

Move the centre and change the scale factor. The dashed rays from the centre pass through matching corners of the object and the image. Try a scale factor of 0.5.

Worked examples

Worked example 1

Drawing an enlargement

OA = 3 cm, OB = 2 cm and OC = 4 cm. Enlarge triangle ABC with centre O and scale factor 2.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Finding the centre and scale factor

Triangle BCD is enlarged to B′C′D′. CB = 3 and C′B′ = 9. How do you describe the enlargement?

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Area after enlargement

A triangle has perimeter 12 cm and area 6 cm². Find the perimeter and area of its image after an enlargement with scale factor 3.

Try it yourself first, then reveal the solution one step at a time.

Practice: enlargement

9 questions

Nine questions on scale factors, centres, perimeter and area.
  1. Question 1 of 9Explore and matchFoundation
    Enlarge the triangle with scale factor 2 and centre .

    Move the controls until you match the question, then press Check.

    Scale factor
    2
    Centre of enlargement
    (1, 1)
    Lengths
    every length is 2 times as long
    Area
    the area is 4 times as big
  2. Question 2 of 9Complete the table
    A triangle has perimeter 12 cm and area 6 cm². Complete the table for its images.
    Scale factorPerimeter (cm)Area (cm²)
    2
    3
    4
  3. Question 3 of 9Multiple choice
    Virenda's trapezium has perimeter 16 cm and area 16 cm². After an enlargement the perimeter is 96 cm. What is the new area?
  4. Question 4 of 9Type the answer
    A pentagon has perimeter 50 cm. Its image has perimeter 300 cm. What is the scale factor?
  5. Question 5 of 9Type the answer
    A hexagon of area 21 cm² is enlarged by scale factor . The image has area 525 cm². Find .
  6. Question 6 of 9Fill the gaps
    Polygon B is an enlargement of polygon A with scale factor 3. B's perimeter is 24 cm and A's area is 7 cm².

    A's perimeter: cm B's area: cm²

  7. Question 7 of 9Type the answer
    Point A is 2 right and 3 up from the centre of enlargement. Where is A′ with scale factor 3? Give how many squares right.
    squares right
  8. Question 8 of 9True or false
    An enlargement with scale factor makes the shape smaller.
  9. Question 9 of 9Multiple choice
    Which of these always gives an image that is congruent (same size and shape) to the object?