Triangle ABC with A(2,3), B(4,3), and C(2,5) is rotated +90° about point K(1,1). What are the coordinates of image vertex A'?

(-1, 2). First, find the relative position vector KA = pmatrix 2-1 \ 3-1 pmatrix = pmatrix 1 \ 2 pmatrix. Rotating by +90° converts pmatrix x \ y pmatrix to pmatrix -y \ x pmatrix, so KA' = pmatrix -2 \ 1 pmatrix. Adding centre K(1,1) gives A' = (1-2, 1+1) = (-1, 2).

Grade 10 · Core Mathematics · Lesson 5 of 13

Rotating Objects on the Cartesian Plane About Any Pivot Point

A Grade 10 Core Mathematics CBE lesson from the sub-strand Rotation. In it, learners locate and mark the centre of rotation and object vertices on a Cartesian plane.

Strand Measurements and GeometrySub-strand RotationLesson code 10CM-23-L540 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

Locate and mark the centre of rotation and object vertices on a Cartesian plane.
Construct image vertices using angle measurements and equal distances from a non-origin pivot.
Calculate exact image coordinates for standard rotations of $90^{\circ}$ and $180^{\circ}$ about any point $(x_{0}, y_{0})$ .

Key vocabulary

Centre of rotation: The fixed pivot point about which a shape is turned on a plane.; Point $C (1, 1)$ acts as the centre of rotation for mapping triangle $P Q R$ .
Angle of rotation: The amount and direction of turn, measured in degrees, about a pivot point.; An angle of rotation of $- 90^{\circ}$ turns the figure clockwise by a quarter turn.
Ray: A line starting at the centre of rotation and extending through a shape's vertex.; Each ray drawn from $C (1, 0)$ through a vertex guides the angle measurement.
Congruence: The property where two figures have identical shape and corresponding equal dimensions.; Rotation preserves congruence so parallelogram $A B C D$ and image $A^{'} B^{'} C^{'} D^{'}$ are congruent.

Lesson preview

Slide 1 · Rotating About Arbitrary Pivot Points on the GridOpening · 12 min

Figure 1: A coordinate grid shows triangle PQR rotated ninety degrees clockwise about the pivot point C.

When performing a rotation on a Cartesian plane where the pivot is not at the origin, we use the centre of rotation as our local reference point. We calculate or measure the direction and distance from this centre to each vertex of the object figure. Applying a turn of $- 90^{\circ}$ rotates every connecting segment clockwise by a right angle. The resulting image vertices maintain equal distances from the pivot point. Use Figure 1 to see how a ninety-degree clockwise rotation changes a figure's position.

To rotate a shape on the Cartesian plane, measure angles from the centre of rotation rather than the origin.
Vector displacements from the centre to each vertex are rotated by the given turn angle.
A negative angle indicates a clockwise turn, while a positive angle indicates an anticlockwise turn.
The distance between the centre of rotation and each image vertex remains equal to the original distance.

Worked example. 1. Find relative position vectors from centre

C (1, 1)

\vec{C P} = (\begin{array}{c} 3 - 1 \\ 2 - 1 \end{array}) = (\begin{array}{c} 2 \\ 1 \end{array})

\vec{C Q} = (\begin{array}{c} 4 \\ 3 \end{array})

\vec{C R} = (\begin{array}{c} 1 \\ 4 \end{array})

. 2. Rotate each vector by

- 90^{\circ}

(clockwise turn:

(\begin{array}{c} x \\ y \end{array}) \to (\begin{array}{c} y \\ - x \end{array})

\vec{C P^{'}} = (\begin{array}{c} 1 \\ - 2 \end{array})

\vec{C Q^{'}} = (\begin{array}{c} 3 \\ - 4 \end{array})

\vec{C R^{'}} = (\begin{array}{c} 4 \\ - 1 \end{array})

. 3. Add centre coordinates

C (1, 1)

to find image positions:

P^{'} = (1 + 1, 1 - 2) = (2, - 1)

Q^{'} = (1 + 3, 1 - 4) = (4, - 3)

R^{'} = (1 + 4, 1 - 1) = (5, 0)

. Answer: $P^{'} (2, - 1)$ , $Q^{'} (4, - 3)$ , $R^{'} (5, 0)$ .

Check question: Triangle $A B C$ with $A (2, 3)$ , $B (4, 3)$ , and $C (2, 5)$ is rotated $+ 90^{\circ}$ about point $K (1, 1)$ . What are the coordinates of image vertex $A^{'}$ ?

$(- 1, 2)$
$(3, 0)$
$(- 3, 2)$
$(- 2, 1)$

Show answer

$(- 1, 2)$ . First, find the relative position vector $\vec{K A} = (\begin{array}{c} 2 - 1 \\ 3 - 1 \end{array}) = (\begin{array}{c} 1 \\ 2 \end{array})$ . Rotating by $+ 90^{\circ}$ converts $(\begin{array}{c} x \\ y \end{array})$ to $(\begin{array}{c} - y \\ x \end{array})$ , so $\vec{K A^{'}} = (\begin{array}{c} - 2 \\ 1 \end{array})$ . Adding centre $K (1, 1)$ gives $A^{'} = (1 - 2, 1 + 1) = (- 1, 2)$ .

Slide 2 · Geometric Construction of Non-Standard RotationsModel · 14 min

Figure 2: The geometric construction of a sixty-degree anticlockwise rotation of a single point around a pivot.

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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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