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Grade 10 · Core Mathematics · Lesson 4 of 13

Rotating Objects on the Cartesian Plane

A Grade 10 Core Mathematics CBE lesson from the sub-strand Rotation. In it, learners rotate a figure on a Cartesian grid about the origin through a specified angle and determine image coordinates.

Strand Measurements and GeometrySub-strand RotationLesson code 10CM-23-L440 minutes · 3 slides

KICD learning outcome (b)

Rotate an object given the centre and angle of rotation on a plane surface and the Cartesian plane.

By the end of this lesson, learners should be able to

  1. Rotate a figure on a Cartesian grid about the origin through a specified angle and determine image coordinates.
  2. Perform rotations about an arbitrary centre point on a Cartesian plane using protractors and equal distance rays.
  3. Compare and record object and image coordinates to verify rotational transformations.

Key vocabulary

Centre of rotation
The fixed point around which a shape turns during a rotational transformation.
The origin O(0,0) acts as the centre of rotation for turning triangle JKL by +90 deg.
Angle of rotation
The amount of angular turn measured in degrees around a fixed centre point.
Rotating triangle ABC through plus ninety degrees turns each segment around pivot M(1,1).
Clockwise
Turning in the same direction as the hands of a clock, indicated by negative angles.
A turn of minus ninety degrees moves points in a clockwise direction around centre N(2,2).

Lesson preview

Slide 1 · Rotating About the Origin on a GridOpening · 12 min
A Cartesian plane showing triangle JKL rotated by positive ninety degrees about the origin to form image triangle J'K'L'.
Figure 1: A Cartesian plane showing triangle JKL rotated by positive ninety degrees about the origin to form image triangle J'K'L'.

To rotate a figure on the Cartesian plane about the origin, connect each vertex to the origin using straight line segments. Measure the required angle from each segment in the specified direction, anticlockwise for positive angles and clockwise for negative angles. Mark each image vertex at an equal distance from the origin along the rotated ray, then connect the image points.

  • Joining each object vertex to the origin forms rays that serve as references for measuring rotation angles.
  • Positive rotation angles turn ray segments in an anticlockwise direction on the Cartesian plane.
  • Each image vertex lies at the same distance from the origin as its original object vertex.
  • Comparing resulting coordinates helps learners confirm patterns for common standard turns like positive ninety degrees.
Worked example. Plot triangle JKL with vertices J(1, 3), K(4, 1), and L(2, 5) on the Cartesian plane. Draw line segment OJ from origin O(0,0) to J(1, 3). Rotate OJ anticlockwise through +90° to locate line segment OJ'. Since OJ has length 10, set OJ' = 10 at (−3,1) to give J'(-3, 1). Repeat the +90° anticlockwise turn for ray OK to find K'(-1, 4) and for ray OL to find L'(-5, 2). Join points J'(-3, 1), K'(-1, 4), and L'(-5, 2) to draw image triangle J'K'L'. Answer: J'(-3, 1), K'(-1, 4), L'(-5, 2).

Check question: Point P has coordinates (2, 4). Find the coordinates of image P' when P is rotated about the origin O(0,0) through +90°.

  1. (-4, 2)
  2. (4, -2)
  3. (-2, -4)
  4. (-4, -2)
Show answer

(-4, 2). A positive 90° rotation means turning anticlockwise. This transformation maps any point (x, y) around the origin to (-y, x). Therefore, the x-coordinate becomes -4 and the y-coordinate becomes 2, so P(2, 4) becomes P'(-4, 2).

Slide 2 · Rotating About an Arbitrary Pivot PointModel · 14 min
A Cartesian grid showing triangle ABC rotated by positive ninety degrees about a pivot point M to form image triangle A'B'C'.
Figure 2: A Cartesian grid showing triangle ABC rotated by positive ninety degrees about a pivot point M to form image triangle A'B'C'.

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

  • All 3 slides with diagrams and class activities
  • Teacher notes, SNE adaptations and assessment checks
  • Homework and the other 12 lessons in this sub-strand
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All 13 lessons in Rotation

  1. 1Properties of Rotation: Centre, Angle, and CongruenceDetermine the properties of rotation in different situations
  2. 2Properties of Rotation and Evaluating Rotational TransformationsDetermine the properties of rotation in different situations
  3. 3Rotating Shapes on a Plane SurfaceRotate an object given the centre and angle of rotation on a plane surface and the Cartesian plane
  4. 4Rotating Objects on the Cartesian PlaneRotate an object given the centre and angle of rotation on a plane surface and the Cartesian plane
  5. 5Rotating Objects on the Cartesian Plane About Any Pivot PointRotate an object given the centre and angle of rotation on a plane surface and the Cartesian plane
  6. 6Finding the Pivot Centre and Angle of TurnDetermine the centre and angle of rotation given an object and its image
  7. 7Determining Centre and Angle of RotationDetermine the centre and angle of rotation given an object and its image
  8. 8Determining the Order of Rotational Symmetry of Plane FiguresDetermine the order of rotational symmetry of plane figures
  9. 9Determining Symmetrical Axes and Rotational Orders in 3D ShapesDetermine the axis and order of rotational symmetry in solids
  10. 10Rotational Symmetry in 3D Solids: Axes and OrdersDetermine the axis and order of rotational symmetry in solids
  11. 11Deducing Congruence from RotationDeduce congruence from rotation
  12. 12Deducing Congruence from Rotation and Identifying TransformationsDeduce congruence from rotation
  13. 13Applications of Rotation and Rotational Symmetry in Real LifeAppreciate the application of rotation in real-life situations