Grade 10 · Core Mathematics · Lesson 12 of 13

Deducing Congruence from Rotation and Identifying Transformations

A Grade 10 Core Mathematics CBE lesson from the sub-strand Rotation. In it, learners deduce that rotation preserves shape, size, and vertex sense to produce directly congruent figures.

Strand Measurements and GeometrySub-strand RotationLesson code 10CM-23-L1240 minutes · 2 slides

KICD learning outcome (f)

Deduce congruence from rotation.

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

Deduce that rotation preserves shape, size, and vertex sense to produce directly congruent figures.
Distinguish direct congruence produced by rotation from opposite congruence produced by reflection.
Determine the transformation type (rotation, translation, or reflection) and calculate its parameters including centre and angle of turn.

Key vocabulary

Direct Congruence: Congruence where the object and image maintain the same order or sense of vertices.; Parallelogram $A B C D$ rotated onto $A^{'} B^{'} C^{'} D^{'}$ preserves its anticlockwise vertex sense, demonstrating direct congruence.
Vertex Sense: The clockwise or anticlockwise order of vertices around a closed geometric figure.; Vertices $J$ , $K$ , $L$ , $M$ listed in order give a clockwise orientation.
Opposite Congruence: Congruence where the image has the reversed vertex sense compared to the original object.; Reflecting a right-angled triangle across an axis produces opposite congruence because vertex direction reverses.
Centre of Rotation: The fixed point about which all points of a figure turn during a rotation.; The intersection of perpendicular bisectors of matching vertices identifies the unique centre of rotation.

Lesson preview

Slide 1 · Rotational Direct Congruence vs Opposite CongruenceModel · 18 min

Figure 1: This diagram compares direct congruence through rotation, which preserves orientation, to opposite congruence through reflection, which reverses it.

Rotation is a rigid motion transformation that preserves side lengths and interior angles, producing congruent figures. When an object is rotated about a fixed pivot centre, its image retains the original orientation sense, meaning vertices listed in order around the shape maintain the same clockwise or anticlockwise direction. This preservation of vertex sense defines direct congruence. In contrast, reflection flips the figure, reversing vertex sense to produce opposite congruence. Therefore, any two shapes related purely by rotation are directly congruent. Use Figure 1 to see how rotation preserves orientation while reflection reverses it.

Rotation preserves both side lengths and angle measures, producing congruent image figures.
Direct congruence occurs when object and image share the same clockwise or anticlockwise vertex sense.
Reflection reverses the vertex sense of a geometric figure, resulting in opposite congruence.
Rotating an irregular quadrilateral or triangle preserves its exact shape, size, and orientation sense.

Worked example. 1. Apply the

90^{\circ}

anticlockwise rotation rule

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

to each vertex. 2. Calculate image vertices:

J^{'} (- 2, 1)

K^{'} (- 5, 3)

L^{'} (- 4, 5)

, and

M^{'} (- 1, 3)

. 3. Trace vertex order

J \to K \to L \to M

(clockwise) and

J^{'} \to K^{'} \to L^{'} \to M^{'}

(clockwise). 4. Since lengths and angles are preserved and vertex sense remains clockwise, the shapes are directly congruent. Answer: Image vertices are $J^{'} (- 2, 1)$ , $K^{'} (- 5, 3)$ , $L^{'} (- 4, 5)$ , $M^{'} (- 1, 3)$ , and $J^{'} K^{'} L^{'} M^{'}$ is directly congruent to $J K L M$ by rotation..

Check question: Triangle $P Q R$ has vertices $P (2, 1)$ , $Q (2, 4)$ , and $R (5, 1)$ with a right angle at $P$ . Its image, triangle $P^{'} Q^{'} R^{'}$ , has vertices $P^{'} (2, - 1)$ , $Q^{'} (2, - 4)$ , and $R^{'} (5, - 1)$ . Is $P^{'} Q^{'} R^{'}$ directly congruent to $P Q R$ by a rotation?

No, because the vertex sense is reversed.
Yes, because their corresponding side lengths are equal.
Yes, because the vertex sense is preserved.
No, because the corresponding side lengths are not equal.

Show answer

No, because the vertex sense is reversed. Tracing $P (2, 1) \to Q (2, 4) \to R (5, 1)$ goes clockwise, while $P^{'} (2, - 1) \to Q^{'} (2, - 4) \to R^{'} (5, - 1)$ goes anticlockwise. A rotation preserves vertex sense, so this transformation is a reflection across the x-axis, giving opposite congruence rather than direct congruence.

Slide 2 · Identifying Transformations and Finding Rotation ParametersPractice · 22 min

Figure 2: This coordinate grid shows two examples of transformations, demonstrating how to identify mappings using vertices and side orientations.

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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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Deducing Congruence from Rotation and Identifying 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