Once you have proved that the translation image of a segment...

Once you have proved that the translation image of a segment is congruent to the preimage, how could you prove the following? Write a plan for each proof.
If point $C$ is between points $A$ and $B,$ then the translation $C$ is between $A^{\prime}$ and $B^{\prime} .$

Key Concepts

Invariance Under Isometries

Invariance under isometries, such as translations, indicates that certain geometric relations and properties (like congruence, collinearity, and betweenness) are unchanged when a figure undergoes a transformation that preserves distances and angles. This fact is crucial in constructing proofs because it shows that if a point has a specific relational property in the original figure, its image under an isometry will have the same property relative to the images of the other points.

Betweenness

Betweenness is a fundamental concept in geometry that determines the order of points along a line. It means that if one point lies between two others, the distance from the first point to the third is the sum of the distances from the first to the middle point and from the middle point to the third. This concept is essential when considering the effects of an isometric transformation like a translation, as it ensures that the linear order of points is maintained.

Congruence Preservation

Congruence preservation refers to the fact that a translation maps a geometric figure onto another figure that is congruent to it, meaning that the shapes are identical in size and form. This implies that any segment, including its length and internal relations like betweenness, remains consistent after the translation, providing a direct basis for proofs involving these properties.

Translation

Translation is a type of isometry where every point in the plane or space is shifted the same distance in a specified direction. This transformation preserves lengths, angles, and the overall configuration of figures. Because of these properties, translations keep the structural relations between points intact, such as collinearity and the order of points along a line.

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