Grade 10 · Core Mathematics · Lesson 7 of 13

Determining Centre and Angle of Rotation

A Grade 10 Core Mathematics CBE lesson from the sub-strand Rotation. In it, learners determine the centre of rotation geometrically using perpendicular bisectors of corresponding vertices.

Strand Measurements and GeometrySub-strand RotationLesson code 10CM-23-L740 minutes · 3 slides

KICD learning outcome (c)

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

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

Determine the centre of rotation geometrically using perpendicular bisectors of corresponding vertices.
Calculate or measure the angle and direction of rotation mapping an object onto its image.
Describe fully a rotation given object and image figures on a coordinate grid.

Key vocabulary

Centre of rotation: The fixed point around which all other points of a figure turn during rotation.; The perpendicular bisectors of segments $E E^{'}$ and $F F^{'}$ intersect at the centre of rotation $O (0, 1)$ .
Angle of rotation: The angle formed between lines from the centre of rotation to matching object-image vertices.; Measuring the turn from $O E$ to $O E^{'}$ gives an angle of rotation of $+ 90^{\circ}$ .
Perpendicular bisector: A line passing through the midpoint of a segment at a right angle.; The perpendicular bisector of segment $A A^{'}$ passes through midpoint $(2, 0)$ with gradient $- \frac{1}{2}$ .
Congruence: The property where two shapes have identical side lengths and angle measures.; Square $A B C D$ and square $A^{'} B^{'} C^{'} D^{'}$ demonstrate congruence under rotational transformation.

Lesson preview

Slide 1 · Constructing the Centre of Rotation GeometricallyModel · 12 min

To locate the centre of rotation $O$ geometrically, join corresponding object and image points with straight line segments, such as $E E^{'}$ and $F F^{'}$ . Construct the perpendicular lines that bisect these segments. The point where these perpendicular bisectors intersect gives the centre of rotation $O$ . Joining $O$ to any object vertex $E$ and its image $E^{'}$ allows us to measure or calculate the directed angle of rotation $∠ E O E^{'}$ .

Straight lines joining corresponding object and image vertices define segments requiring perpendicular bisectors.
The point of intersection of any two perpendicular bisectors determines the centre of rotation $O$ .
Joining the centre $O$ to corresponding vertices forms the angle of rotation, measured from object to image.
An anticlockwise turn is written as a positive angle, while a clockwise turn is negative.

Worked example. Find the perpendicular bisector of

E E^{'}

: midpoint is

(0, 2)

and segment is horizontal, so the perpendicular bisector is the vertical line

x = 0

. Find the perpendicular bisector of

F F^{'}

: midpoint is

(1.5, 3.5)

and slope of

F F^{'}

- \frac{3}{5}

, so perpendicular slope is

\frac{5}{3}

. Equation of second bisector is

y - 3.5 = \frac{5}{3} (x - 1.5)

, simplifying to

y = \frac{5}{3} x + 1

. Find intersection

O

: substitute

x = 0

into

y = \frac{5}{3} x + 1

to get

y = 1

, giving centre

O (0, 1)

. Calculate angle

∠ E O E^{'}

: vector

O E = (1, 1)

and vector

O E^{'} = (- 1, 1)

, which are perpendicular, giving an anticlockwise turn of

+ 90^{\circ}

. Answer: Centre of rotation is $O (0, 1)$ and angle of rotation is $+ 90^{\circ}$ ..

Check question: The segments connecting an object to its image, $A A^{'}$ and $B B^{'}$ , have perpendicular bisectors given by the equations $x = 2$ and $y = 3$ . What are the coordinates of the centre of rotation $O$ ?

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

Show answer

$(2, 3)$ . The centre of rotation is the unique point where the perpendicular bisectors intersect. The line $x = 2$ and the line $y = 3$ intersect at the point with an x-coordinate of $2$ and a y-coordinate of $3$ , which gives the point $(2, 3)$ .

Slide 2 · Finding Rotation Parameters from Coordinate VerticesPractice · 14 min

Figure 1: A coordinate plane showing triangle ABC and its rotated image A'B'C', with lines connecting corresponding vertices intersecting at the centre O(2,0).

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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