The given transformation of reflection, rotation and translation are rigid transformation
The transformation that must take any point A to any point B are;
- <u>Rotation of 180° around the midpoint of segment AB</u>
- <u>Reflection across the perpendicular bisector</u>
- <u>Translation by the directed line segment AB</u>
Reason:
<em>Question: The given quadrilaterals ABCD and A'B'C'D' are congruent</em>
Required; To select all transformation that must take a point A to point B
Solution:
- Rotation of 180° around the midpoint of segment AB
A rotation of 180° about the midpoint of segment AB will change the places of points A and B
A(x, y) → R₁₈₀ → A'(-x, -y)
Let the coordinates of the origin be the midpoint of segment AB, and let (x, y) represent the coordinate of the point B(x, 0) we have;
∴ The coordinates of point A is A(-x, 0), which gives;
A(-x, 0) → R₁₈₀ → A'(x, 0) = point B
Therefore a rotation about the midpoint of AB must take point A to point B
- Reflection across the perpendicular bisector
The reflection of a point (x, y) about y-axis gives a point (-x, y), therefore, we have;
A(-x, 0) → Reflection across the y-axis → A'(x, 0) = point B
Therefore, a reflection across the perpendicular bisector of the line AB must take the point A to point B
- Translation by the directed line segment AB
A translation along the directed line segment AB of point A(-x, 0) 2·x units right gives;
A'(-x + 2·x, 0) = A'(x, 0) = Point B
Therefore, the transformations that must take any point A to any point B are;
<u>Rotation of 180° around the midpoint of segment AB</u>
<u>Reflection across the perpendicular bisector</u>
<u>Translation by the directed line segment AB</u>
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