Question

anna tried to define a translation.

• point ( a ) maps to point ( a ).
• every point ( p ) maps to point ( p ) such that ( overrightarrow{pp} ) is parallel to ( overrightarrow{aa} ).
• ( overline{pp} ) has the same length as ( overline{aa} ).

which counterexample shows that anna’s definition does not fully define a translation?
choose 1 answer:
(options with images of point mappings for ( a \to a ) and ( p \to p ) as described in the problem, and option d: anna fully defined a translation)

Explanation:

A translation requires that the vector \(\overrightarrow{PP'}\) has the same direction as \(\overrightarrow{AA'}\), not just the same length and parallel. In option C, the direction of \(\overrightarrow{PP'}\) is opposite to \(\overrightarrow{AA'}\) (since \(P'\) is to the left of \(P\) while \(A'\) is to the right of \(A\)), but it still satisfies being parallel and same length. This shows Anna's definition is incomplete because it doesn't account for direction, so C is the counterexample.

Answer:

C. The diagram where \(A\) maps to \(A'\) (rightward) and \(P'\) maps to \(P\) (leftward, so \(\overrightarrow{PP'}\) is opposite in direction to \(\overrightarrow{AA'}\) but parallel and same length)