Grade 10 · Core Mathematics · Lesson 2 of 13

Properties of Rotation and Evaluating Rotational Transformations

A Grade 10 Core Mathematics CBE lesson from the sub-strand Rotation. In it, learners verify that the distance from the centre of rotation to any object point equals the distance to its corresponding image point.

Strand Measurements and GeometrySub-strand RotationLesson code 10CM-23-L240 minutes · 2 slides

KICD learning outcome (a)

Determine the properties of rotation in different situations.

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

Verify that the distance from the centre of rotation to any object point equals the distance to its corresponding image point.
Confirm that all points on a rotated object rotate through an identical angle about the centre.
Evaluate whether a given transformation between two figures is a rotation about a specified pivot point.

Key vocabulary

Centre of rotation: The fixed point about which a figure turns during a rotational transformation.; The origin $O (0, 0)$ acts as the centre of rotation for triangle $L M N$ .
Invariant point: A point that does not change position under a given geometric transformation.; The pivot centre $O (0, 0)$ is the only invariant point during the rotation.
Angle of rotation: The amount of turn in degrees through which a figure is rotated about a centre.; The angle of rotation from quadrilateral ABCD to A'B'C'D' about K(0,1) is 180 deg.
Congruence: The state of figures having identical size and shape with equal corresponding side lengths.; Quadrilateral ABCD and its rotated image A'B'C'D' demonstrate complete congruence on the grid.

Lesson preview

Slide 1 · Core Properties of RotationOpening · 18 min

Figure 1: A coordinate grid showing triangle LMN rotated 90 degrees anticlockwise about the origin to form image triangle L'M'N'.

A rotation is a transformation determined by a fixed pivot centre and a specific angle of turn. Every point on the object and its image lie at equal distances from the pivot centre. Furthermore, every line connecting an object point to the centre rotates through the exact same angle to reach its image point. Anticlockwise turns are defined as positive, while clockwise turns are negative.

A geometric rotation is fully determined by a fixed pivot centre and a specified angle of turn.
Each object point and its corresponding image point lie at equal distances from the pivot centre.
The angle formed between any object point, the pivot centre, and its image point remains constant throughout.
Anticlockwise rotations are assigned positive angle measures, whereas clockwise rotations are assigned negative angle measures.

Worked example. Apply the

+ 90^{\circ}

anticlockwise rotation rule

(x, y) \to (- y, x)

about origin

O (0, 0)

. Calculate coordinates:

L^{'} (- 1, 2)

M^{'} (- 1, 5)

, and

N^{'} (- 4, 2)

. Calculate object distance

O M = \sqrt{5^{2} + 1^{2}} = \sqrt{26} \approx 5.10

units. Calculate image distance

O M^{'} = \sqrt{(- 1)^{2} + 5^{2}} = \sqrt{26} \approx 5.10

units, confirming

O M = O M^{'}

. Answer: Image coordinates are $L^{'} (- 1, 2)$ , $M^{'} (- 1, 5)$ , $N^{'} (- 4, 2)$ , and distances $O M = O M^{'} = \sqrt{26} \approx 5.10 units$ ..

Check question: A point $P (3, - 2)$ is rotated $90^{\circ}$ clockwise about the origin $O (0, 0)$ to point $P^{'}$ . State the coordinates of $P^{'}$ and calculate distance $O P$ .

$P^{'} (- 2, - 3)$ and $O P = \sqrt{13} \approx 3.61 units$
$P^{'} (2, 3)$ and $O P = \sqrt{13} \approx 3.61 units$
$P^{'} (- 2, - 3)$ and $O P = \sqrt{5} \approx 2.24 units$
$P^{'} (2, 3)$ and $O P = \sqrt{5} \approx 2.24 units$

Show answer

$P^{'} (- 2, - 3)$ and $O P = \sqrt{13} \approx 3.61 units$ . Rotating $P (3, - 2)$ by $90^{\circ}$ clockwise uses the rule $(x, y) \to (y, - x)$ , giving $P^{'} (- 2, - 3)$ . The distance from the origin is $O P = \sqrt{3^{2} + (- 2)^{2}} = \sqrt{9 + 4} = \sqrt{13} \approx 3.61 units$ .

Slide 2 · Evaluating Rotations and Finding Turning AnglesPractice · 22 min

Figure 2: A coordinate plane displaying quadrilateral ABCD rotated 180 degrees about the pivot point K to form its image A'B'C'D'.

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Both 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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Properties of Rotation and Evaluating Rotational Transformations

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