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Geometric transformations (rotation) — KCSE Mathematics

KCSE Mathematics · 90 practice questions · 3 syllabus objectives · 3 revision lessons

25 easy33 medium32 hard

Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

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

Rotate a point or shape through a given angle (90°, 180°, 270°) about a given centre of rotation

Determine the centre, angle and direction of rotation given an object and its image

Geometric transformations (rotation)

Revision Notes

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

Understanding Rotations in Geometry

In geometry, rotation involves turning a shape or point around a fixed point known as the centre of rotation. The angle of rotation can be 90°, 180°, or 270°.

Steps to Rotate a Point:

  1. Identify the centre of rotation: This is the point around which you will rotate.
  2. Determine the angle of rotation: Know whether you’re rotating clockwise or counterclockwise.
  3. Apply the rotation: Use the following rules for specific angles:
    • 90° clockwise: (x, y) becomes (y, -x)
    • 180°: (x, y) becomes (-x, -y)
    • 270° clockwise: (x, y) becomes (-y, x)

For example, to rotate the point (2, 3) 90° clockwise about the origin (0, 0):

  • Apply the rule: (2, 3) → (3, -2)
  • Thus, the new coordinates are (3, -2).

Key points to remember

  • Rotation turns shapes around a fixed point.
  • Identify the centre of rotation before rotating.
  • Apply specific rules for 90°, 180°, and 270° rotations.
  • Clockwise and counterclockwise directions matter.
  • Practice with different shapes for mastery.

Worked example

Rotate the point (1, 2) 180° about the origin.

  • Applying the rule: (1, 2) → (-1, -2).
  • The new coordinates are (-1, -2).

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More lessons in this topic

Lesson 2: Understanding Rotation in Geometry

Objective: Determine the centre, angle and direction of rotation given an object and its image

Rotation is a type of geometric transformation that turns a shape around a fixed point called the centre of rotation. To determine the centre, angle, and direction of rotation, follow these steps:

  1. Identify the Centre of Rotation: This is the point that remains fixed while the object rotates. You can find it by drawing lines from corresponding points on the object to the image. The intersection of these lines is the centre.

  2. Determine the Angle of Rotation: Measure the angle formed between the original position of a point and its new position after rotation. Use a protractor for accuracy.

  3. Establish the Direction of Rotation: This can be either clockwise or counterclockwise. A clockwise rotation moves points in the direction of a clock's hands, while counterclockwise moves them in the opposite direction.

For example, if point A (2, 3) rotates to point A' (3, 2) around point O (1, 1), the steps are:

  • Draw OA and OA'.
  • The angle ∠AOB is 90°.
  • The rotation is counterclockwise.

By following these steps, you can effectively determine the parameters of rotation for any geometric figure.

  • Centre of rotation is the fixed point during rotation.
  • Angle of rotation is measured between original and new position.
  • Direction can be clockwise or counterclockwise.
  • Use corresponding points to find the centre.
  • Protractors help in measuring angles accurately.

Given a triangle ABC rotates to A'B'C' around point O. If A(2, 3) moves to A'(3, 2), the angle is 90° counterclockwise.

Lesson 3: Understanding Rotations in Geometry

Objective: Geometric transformations (rotation)

Rotation is a type of geometric transformation that turns a figure around a fixed point called the center of rotation. The angle of rotation is measured in degrees, and it can be clockwise or counterclockwise. To perform a rotation, follow these steps:

  1. Identify the center of rotation: This is the point around which the figure will rotate.
  2. Determine the angle of rotation: This tells you how far to turn the figure.
  3. Rotate each point of the figure: Use the angle to find the new position of each point.

For example, if you rotate a triangle 90 degrees counterclockwise about the origin, each vertex moves to a new location based on the rotation rule. The coordinates of a point (x, y) after a 90-degree counterclockwise rotation become (-y, x).

Understanding rotations helps in visualizing and solving problems related to symmetry, congruence, and transformations in geometry.

  • Rotation turns a figure around a fixed point.
  • The center of rotation is crucial for transformations.
  • Angles of rotation can be clockwise or counterclockwise.
  • Use rotation rules to find new coordinates.
  • Rotations affect the position but not the size of figures.

Question: Rotate the point (2, 3) 90 degrees counterclockwise about the origin. Answer: The new coordinates are (-3, 2) after applying the rotation rule.

Sample Questions

Read 3 questions and answers free. Sign up to access all 90 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER3 marks

State how to determine the centre of rotation when a quadrilateral PQRS with vertices P(1, 1), Q(3, 1), R(3, 3), and S(1, 3) is rotated to form quadrilateral P'Q'R'S' with vertices P'(1, 3), Q'(3, 3), R'(3, 1), and S'(1, 1). (3 marks)

Answer & marking scheme

Part (a) — 3 marks
Join points P and P' and construct the perpendicular bisector (1 mk)
Join points Q and Q' and construct the perpendicular bisector (1 mk)
The intersection of the two bisectors gives the centre of rotation (1 mk)
2
easySHORT ANSWER4 marks

State the centre of rotation and the angle of rotation when triangle ABC with vertices A(2, 3), B(4, 5), and C(6, 3) is rotated to form triangle A'B'C' with vertices A'(3, 2), B'(5, 4), and C'(3, 6). (4 marks)

Answer & marking scheme

Part (a) — 4 marks
Centre of rotation is (4, 4) (1 mk)
Angle of rotation is 90° (1 mk)
Direction of rotation is clockwise (1 mk)
Coordinates of A'B'C' confirm the transformation (1 mk)
3
easySHORT ANSWER2 marks

A point B(3, 4) is rotated about the origin through an angle of +180°. State the coordinates of the image B′. (2 marks)

Answer & marking scheme

Part (a) — 2 marks
For +180° about the origin: (x, y) → (−x, −y) (1 mk)
B′ = (−3, −4) (1 mk)
4

Identify the coordinates of the image of triangle ABC with vertices A(1, 1), B(2, 3), and C(3, 1) after a rotation of 180° about the origin. (3 marks)

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Frequently asked questions

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

Geometric transformations (rotation) covers Rotate a point or shape through a given angle (90°, 180°, 270°) about a given centre of rotation; Determine the centre, angle and direction of rotation given an object and its image; Geometric transformations (rotation), all aligned to the official KNEC KCSE Mathematics syllabus.

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

HighMarks has 90 Geometric transformations (rotation) 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 (rotation) for the KCSE exam?

Start with the revision notes on this page to refresh the core concepts, then work through the practice questions in increasing difficulty. Sign up for HighMarks to get a personalised study plan that adapts to the topics you keep getting wrong, plus mock exams, subject-wide practice, and detailed performance tracking. See pricing.

Why Practise Geometric transformations (rotation)?

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