Question

2. Select all transformations that must take any point A to any point B.

Answer 1

The given transformation of reflection, rotation and translation are rigid transformation

The transformation that must take any point A to any point B are;

• Rotation of 180° around the midpoint of segment AB

• Reflection across the perpendicular bisector

• Translation by the directed line segment AB

Reason:

Question: The given quadrilaterals ABCD and A'B'C'D' are congruent

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;

Rotation of 180° around the midpoint of segment AB

Reflection across the perpendicular bisector

Translation by the directed line segment AB

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