Subsection 2.3.5 Rotation and Congruence
Teacher Resource 2.3.33.
To assist your teaching, we have prepared lesson resources, aligned with this textbook and the CBC. The Lesson Plan links to syllabus learning outcomes and provides suggest time allocations. The Step-by-Step Guide provides more detailed guidance on how to teach the content, including suggested questions to ask learners, and possible answers.
Learner Experience 2.3.9.
Work in groups
Copy the trangle \(ABC\) and the point \(D\) on a graph paper. Using a ruler and a protractor, rotate the triangle \(-90^\circ\) about point \(D.\)
Place a protractor at the line \(AD\) with the centre of the protractor at \(D\) and measure \(90^\circ.\) Using a ruler draw \(DA'\) such that \(AD\,=\,DA'.\)
Repeat the step above for for vertices \(B \text{ and} C.\)
Exploration 2.3.10. Rotation and Congruence.
Instructions.
Use this interactive board to explore rotations around point \(D\text{:}\)
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Adjust the Angle: Drag the slider at the top left to change the rotation angle. Notice that positive angles rotate the triangle anticlockwise, while negative angles rotate it clockwise.
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Move the Centre: Drag the red point \(D\) to see how changing the centre of rotation affects the final position of the image.
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Observe Congruence: No matter the angle or centre of rotation, notice that the size and shape of the green triangle always exactly match the blue triangle.
Key Takeaway 2.3.35.
Congruence refers to a relationship between two figures or objects, whereby, they are identical in size and shape.
Rotation is a type of transformation that repositions an object but preserves the shape and size of the object. Thus, rotation produces congruent figures.
\(\Delta ABC \) and \(\Delta A'B'C' \) are similar in size and shape. Therefore, they are said to be directly congruent.
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Rotation is a rigid transformation — it preserves distances and angles. Therefore, the image is always congruent to the object.
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Direct congruence: Rotation preserves orientation (unlike reflection, which reverses it).
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Proving congruence after rotation: Calculate all side lengths using the distance formula, verify that corresponding sides are equal, and use SSS/SAS criteria.
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Rotation about any point preserves congruence (not just the origin).
Example 2.3.36.
Triangle \(ABC\) is mapped onto \(A'B'C'\) after a rotation of \(-45^\circ\) and centre of rotation \(D.\)
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\(\Delta ABC \text{ and } \Delta A'B'C'\) have the same shape and size.
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The length of the corresponding sides of \(\Delta ABC \text{ and } \Delta A'B'C'\) are the same.
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Every corresponding internal angle for the triangles remain the same.
Therefore, \(\Delta ABC \text{ and } \Delta A'B'C'\) are said to be directly congruent.
Checkpoint 2.3.37.
Exercises Exercises
1.
Identify the axes of rotational symmetry and their respective order in the following:
2.
Identify the axes of rotational symmetry and their respective order for the following figures:
