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Geometric transformations (enlargement) — KCSE Mathematics

KCSE Mathematics · 86 practice questions · 3 syllabus objectives · 3 revision lessons

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Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Enlarge a shape by a given scale factor from a given centre of enlargement; find the image coordinates

Calculate the scale factor and locate the centre of enlargement given an object and its image; state the ratio of areas

Geometric transformations (enlargement)

Revision Notes

Concise lesson notes for Geometric transformations (enlargement), written to the KCSE Mathematics marking standard. Read the first lesson free below.

Understanding Enlargements in Geometry

Enlargement is a geometric transformation that increases the size of a shape from a specific point called the centre of enlargement. To enlarge a shape by a given scale factor, follow these steps:

  1. Identify the centre of enlargement (point O) and the scale factor (k).
  2. Draw lines from the centre of enlargement to each vertex of the original shape.
  3. Multiply the distance from the centre of enlargement to each vertex by the scale factor to find the new coordinates.
  4. Plot the new points to form the enlarged shape.

For example, if we have a triangle with vertices A(2, 3), B(4, 5), and C(6, 2) and we want to enlarge it by a scale factor of 2 from the centre O(1, 1):

  • For point A:
    Distance OA = √((2-1)² + (3-1)²) = √5
    New distance = 2 * √5
    New coordinates for A' = (1 + 2*(2-1), 1 + 2*(3-1)) = (3, 5)
  • Repeat for points B and C to find B' and C'.

The new coordinates after enlargement will be A'(3, 5), B'(6, 9), and C'(9, 3).

Key points to remember

  • Enlargement increases the size of a shape from a centre.
  • Identify the scale factor and centre of enlargement.
  • Multiply distances from the centre by the scale factor.
  • Plot new coordinates to form the enlarged shape.

Worked example

Enlarge the point P(3, 4) by a scale factor of 3 from O(1, 1).
New coordinates P' = (1 + 3*(3-1), 1 + 3*(4-1)) = (7, 11).

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More lessons in this topic

Lesson 2: Understanding Enlargement in Geometry

Objective: Calculate the scale factor and locate the centre of enlargement given an object and its image; state the ratio of areas

Enlargement is a geometric transformation that increases the size of a shape while maintaining its proportions. To calculate the scale factor, you can use the formula:

Scale Factor (k) = Length of Image / Length of Object

To locate the centre of enlargement, draw lines from corresponding points on the object to the image. The point where these lines intersect is the centre of enlargement.

When you enlarge a shape, the ratio of the areas of the image to the object is the square of the scale factor. For example, if the scale factor is 3, the ratio of areas is 3² = 9.

Example:

  • Given a triangle with vertices A(1, 2), B(2, 3), and C(3, 1) enlarged to A'(3, 6), B'(6, 9), and C'(9, 3).
  • Calculate the scale factor:
    • Length of AB = √((2-1)² + (3-2)²) = √2
    • Length of A'B' = √((6-3)² + (9-6)²) = √18
    • Scale Factor = √18 / √2 = 3
  • The centre of enlargement is found by drawing lines from A to A', B to B', and C to C'.

Thus, the scale factor is 3, and the ratio of areas is 9.

  • Scale factor calculated using length of image over length of object.
  • Centre of enlargement found by intersecting lines from corresponding points.
  • Ratio of areas is the square of the scale factor.
  • Enlargement maintains shape proportions.
  • Draw accurate diagrams for clarity.

Calculate the scale factor if a rectangle with dimensions 2 cm by 3 cm enlarges to 4 cm by 6 cm.

  • Scale Factor = (4 cm / 2 cm) = 2.
  • Ratio of areas = 2² = 4.
Lesson 3: Understanding Enlargement in Geometry

Objective: Geometric transformations (enlargement)

Enlargement is a type of geometric transformation that increases the size of a shape while maintaining its proportions. To describe enlargement, you need to understand the following key concepts:

  • Centre of Enlargement: This is the point from which the shape is enlarged. All points in the shape move away from or towards this point.
  • Scale Factor: This is the ratio that describes how much larger or smaller the new shape will be compared to the original shape. A scale factor greater than 1 indicates enlargement, while a scale factor between 0 and 1 indicates reduction.

To perform an enlargement:

  1. Identify the centre of enlargement.
  2. Determine the scale factor.
  3. Multiply the coordinates of each vertex of the original shape by the scale factor.

For example, if you have a triangle with vertices A(1, 2), B(3, 4), and C(5, 1), and you want to enlarge it by a scale factor of 2 from the origin (0, 0):

  • New A = (1×2, 2×2) = (2, 4)
  • New B = (3×2, 4×2) = (6, 8)
  • New C = (5×2, 1×2) = (10, 2) Thus, the enlarged triangle will have vertices at (2, 4), (6, 8), and (10, 2).
  • Enlargement maintains shape proportions.
  • Identify the centre of enlargement.
  • Calculate new coordinates using scale factor.
  • A scale factor > 1 indicates enlargement.
  • All points move away from the centre.

Enlarge triangle with vertices A(2, 3), B(4, 5) by scale factor 3 from centre (0,0). New A = (2×3, 3×3) = (6, 9); New B = (4×3, 5×3) = (12, 15).

Sample Questions

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1
easySHORT ANSWER4 marks

A square has vertices A(1, 1), B(1, 3), C(3, 3), and D(3, 1). It is enlarged by a scale factor of 0.5 about the point (2, 2). (a) State the coordinates of the image of point A'. (b) Find the coordinates of point B' after the enlargement. (c) What are the coordinates of point D' after the enlargement? (4 marks)

Answer & marking scheme

Part (a) — 1 mark
A' = (2 + 0.5(1 - 2), 2 + 0.5(1 - 2)) = (1.5, 1.5) (1 mk)
Part (b) — 1 mark
B' = (2 + 0.5(1 - 2), 2 + 0.5(3 - 2)) = (1.5, 2.5) (1 mk)
Part (c) — 2 marks
D' = (2 + 0.5(3 - 2), 2 + 0.5(1 - 2)) = (2.5, 1.5) (2 mks)
2
easySHORT ANSWER4 marks

A triangle has vertices P(3, 2), Q(5, 4), and R(4, 1). It is enlarged by a scale factor of 2 about the point (2, 2). (a) State the coordinates of the image of point P'. (b) Find the coordinates of point Q' after the enlargement. (c) What are the coordinates of point R' after the enlargement? (4 marks)

Answer & marking scheme

Part (a) — 1 mark
P' = (2 + 2(3 - 2), 2 + 2(2 - 2)) = (4, 2) (1 mk)
Part (b) — 1 mark
Q' = (2 + 2(5 - 2), 2 + 2(4 - 2)) = (8, 6) (1 mk)
Part (c) — 2 marks
R' = (2 + 2(4 - 2), 2 + 2(1 - 2)) = (6, 0) (2 mks)
3
easySHORT ANSWER4 marks

A pentagon has vertices A(0, 0), B(2, 0), C(2, 2), D(1, 3), and E(0, 2). It is enlarged by a scale factor of 1.5 about the point (1, 1). (a) State the coordinates of the image of vertex A'. (b) Find the coordinates of vertex B' after the enlargement. (c) What are the coordinates of vertex D' after the enlargement? (4 marks)

Answer & marking scheme

Part (a) — 1 mark
A' = (1 + 1.5(0 - 1), 1 + 1.5(0 - 1)) = (-0.5, -0.5) (1 mk)
Part (b) — 1 mark
B' = (1 + 1.5(2 - 1), 1 + 1.5(0 - 1)) = (2.5, -0.5) (1 mk)
Part (c) — 2 marks
D' = (1 + 1.5(1 - 1), 1 + 1.5(3 - 1)) = (1, 4) (2 mks)
4

A rectangle has vertices A(1, 2), B(3, 2), C(3, 4), and D(1, 4). It is enlarged by a scale factor of 2 about the origin. (a) State the new coordinates of vertex A'. (b) What are the coordinates of vertex B' after the enlargement? (c) Find the coordinates of vertex C' after the enlargement. (4 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Geometric transformations (enlargement)" cover?

Geometric transformations (enlargement) covers Enlarge a shape by a given scale factor from a given centre of enlargement; find the image coordinates; Calculate the scale factor and locate the centre of enlargement given an object and its image; state the ratio of areas; Geometric transformations (enlargement), all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Geometric transformations (enlargement)?

HighMarks has 86 Geometric transformations (enlargement) practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Geometric transformations (enlargement) for the KCSE exam?

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