Grade 10 · Core Mathematics · Lesson 6 of 13

Finding the Pivot Centre and Angle of Turn

A Grade 10 Core Mathematics CBE lesson from the sub-strand Rotation. In it, learners find the centre of rotation by constructing perpendicular bisectors of lines joining matching object and image points.

Strand Measurements and GeometrySub-strand RotationLesson code 10CM-23-L640 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

Find the centre of rotation by constructing perpendicular bisectors of lines joining matching object and image points.
Determine the magnitude and direction of the angle of turn about the identified centre.

Key vocabulary

Perpendicular bisector: A line dividing a given line segment into two equal halves at right angles.; The perpendicular bisector of line segment $P P^{'}$ passes through the centre of rotation $O$ .
Angle of rotation: The measure of angular turn about the centre mapping an object to image.; Joining $A (2, 4)$ and $A^{'} (- 4, 2)$ to $O (0, 0)$ gives an angle of turn of $+ 90^{\circ}$ .
Centre of rotation: The fixed pivot point around which a plane figure is rotated.; The centre of rotation for mapping triangle $E F G$ to $E^{'} F^{'} G^{'}$ is $O (0, 0)$ .

Lesson preview

Slide 1 · Locating the Rotation Centre Using Perpendicular BisectorsOpening · 12 min

Figure 1: A Cartesian plane displaying object points P and Q, their images P' and Q', and the perpendicular bisectors that intersect at the rotation centre.

When an object figure undergoes rotation onto an image figure, every point and its matching image vertex are equidistant from the fixed centre of rotation. Consequently, the rotation centre lies on the perpendicular bisector of line segments joining paired points. By constructing perpendicular bisectors for at least two pairs of matching object and image vertices, their intersection point uniquely identifies the centre of rotation $O$ .

Joining paired object and image vertices forms line segments that share a common perpendicular bisector intersection.
The perpendicular bisector of any segment $A A^{'}$ contains all points equidistant from $A$ and $A^{'}$ .
Constructing perpendicular bisectors for two independent pairs of matching points pinpoints the centre of rotation $O$ .
Connecting centre $O$ to matching vertices allows verification of equal radial distances and rotation angles.

Worked example. Find midpoint of

P P^{'}

(\frac{1 + (- 1)}{2}, \frac{3 + 1}{2}) = (0, 2)

. Gradient of

P P^{'} = \frac{1 - 3}{- 1 - 1} = 1

, so perpendicular gradient =

- 1

. Line equation:

y = - x + 2

. Find midpoint of

Q Q^{'}

(\frac{4 + 1}{2}, \frac{1 + 4}{2}) = (2.5, 2.5)

. Gradient of

Q Q^{'} = \frac{4 - 1}{1 - 4} = - 1

, so perpendicular gradient =

+ 1

. Line equation:

y = x

. Solve simultaneous equations

y = - x + 2

and

y = x

x = - x + 2 ⟹ 2 x = 2 ⟹ x = 1

y = 1

. Answer: The centre of rotation $O$ is $(1, 1)$ ..

Check question: Object point $M (2, 5)$ is mapped to image point $M^{'} (6, 1)$ under a rotation. What is the equation of the line on which the centre of rotation must lie?

$y = x - 1$
$y = - x + 7$
$y = x + 3$
$y = x - 4$

Show answer

$y = x - 1$ . The centre of rotation lies on the perpendicular bisector of $M M^{'}$ . The midpoint of $M M^{'}$ is $(\frac{2 + 6}{2}, \frac{5 + 1}{2}) = (4, 3)$ . The gradient of the segment $M M^{'}$ is $\frac{1 - 5}{6 - 2} = \frac{- 4}{4} = - 1$ . The perpendicular bisector has a gradient that is the negative reciprocal of $- 1$ , which is $m = 1$ . Using the midpoint $(4, 3)$ , the equation of the perpendicular bisector is $y - 3 = 1 (x - 4) ⟹ y = x - 1$ .

Slide 2 · Determining Centre and Angle of Rotation on the Cartesian PlaneModel · 14 min

Figure 2: A coordinate grid illustrating triangle ABC and its image A'B'C' with a measured 90-degree anticlockwise rotation angle about the origin.

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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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Finding the Pivot Centre and Angle of Turn

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

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 13 lessons in Rotation