Grade 10 · Core Mathematics · Lesson 11 of 13

Deducing Congruence from Rotation

A Grade 10 Core Mathematics CBE lesson from the sub-strand Rotation. In it, learners demonstrate that rotation preserves shape, size, and sense of orientation of geometric figures.

Strand Measurements and GeometrySub-strand RotationLesson code 10CM-23-L1140 minutes · 2 slides

KICD learning outcome (f)

Deduce congruence from rotation.

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

Demonstrate that rotation preserves shape, size, and sense of orientation of geometric figures.
Deduce direct congruence between an object and its image under rotational transformations.

Key vocabulary

Direct congruence: Geometric relationship where image matches object in size, shape, and sense of orientation.; Triangle L'M'N' shows direct congruence to triangle LMN after rotation about origin O.
Sense of orientation: The order in which corresponding vertices appear when moving clockwise or anticlockwise around a shape.; Both object triangle LMN and image triangle L'M'N' maintain an anticlockwise vertex order.
Opposite congruence: Geometric relationship where image matches size and shape but has reversed vertex orientation.; A reflected quadrilateral shows opposite congruence because its vertex orientation is flipped.

Lesson preview

Slide 1 · Properties of Rotational Mapping and Direct CongruenceOpening · 18 min

Figure 1: A coordinate grid showing triangle LMN and its image L'M'N' after a ninety-degree anticlockwise rotation about the origin.

When an object undergoes a rotation about a centre point $O$ , every point turns through the exact same angle and direction. This rigid movement preserves all line lengths and interior angles, ensuring the object and its image have identical size and shape. Furthermore, rotation preserves the sense of orientation: tracing vertices in order around the object and image moves in the same direction. Therefore, rotation produces direct congruence between an object and its image.

Rotation maps an object onto an image while preserving all side lengths and interior angle measurements.
Because size and shape are strictly preserved, any rotated figure is congruent to its original object.
Rotation maintains the sense of orientation, meaning vertex order direction remains unchanged.
Figures related by rotation are directly congruent because they can align without flipping across a line.

Worked example. Apply the

90^{\circ}

anticlockwise rotation rule

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

to obtain

L^{'} (- 1, 2)

M^{'} (- 1, 5)

, and

N^{'} (- 5, 2)

. Calculate length

L M = \sqrt{(5 - 2)^{2} + (1 - 1)^{2}} = 3

units and

L^{'} M^{'} = \sqrt{(- 1 - (- 1))^{2} + (5 - 2)^{2}} = 3

units. Observe that all corresponding side lengths and interior angles are preserved, so triangle

L M N

and triangle

L^{'} M^{'} N^{'}

have identical size and shape. The vertex order

L \to M \to N

is anticlockwise in both figures, so the sense is preserved and they are directly congruent. Answer: $L^{'} (- 1, 2)$ , $M^{'} (- 1, 5)$ , $N^{'} (- 5, 2)$ ; side lengths $L M = L^{'} M^{'} = 3$ ; the triangles are directly congruent because size, shape, and sense are preserved..

Check question: Triangle $P Q R$ with vertices $P (0, 3)$ , $Q (4, 0)$ , and $R (0, 0)$ is rotated $180^{\circ}$ about the origin $O (0, 0)$ to form image triangle $P^{'} Q^{'} R^{'}$ . What are the coordinates of $P^{'}$ , and what type of congruence exists between the two triangles?

$P^{'} (0, - 3)$ ; directly congruent
$P^{'} (0, - 3)$ ; oppositely congruent
$P^{'} (- 3, 0)$ ; directly congruent
$P^{'} (- 3, 0)$ ; oppositely congruent

Show answer

$P^{'} (0, - 3)$ ; directly congruent. A $180^{\circ}$ rotation about the origin uses the rule $(x, y) \to (- x, - y)$ , mapping $P (0, 3)$ to $P^{'} (0, - 3)$ . Because a rotation preserves side lengths, interior angles, and the sense of orientation (vertex order), the resulting shapes are directly congruent.

Slide 2 · Distinguishing Rotational Congruence in QuadrilateralsClosing · 22 min

Figure 2: A comparison of two quadrilaterals showing direct congruence from rotation on the left and opposite congruence from reflection on the right.

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.

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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