Question

Give two different sequences of three transformations that would map PQR onto EFG given that PQR=EFG.

15 points!

Answer 1

Translate P to E; rotate ∆PQR about E until Q is coincident with F; reflect ∆PQR across EF

Reflect ∆PQR across line PR; translate R to G; rotate ∆PQR about G until P is coincident with E

Explanation:

The orientations of the triangles are opposite, so a reflection is involved. The various segments are not at right angles to each other, so a rotation other than some multiple of 90° is involved. A translation is needed in order to align the vertices on top of one another.

The rotation is more easily defined if one of the ∆PQR vertices is already on top of its corresponding ∆EFG vertex, so that translation should precede the rotation. The reflection can come anywhere in the sequence.

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Additional comment

The mapping can be done in two transformations: translate a ∆PQR vertex to its corresponding ∆EFG point; reflect across the line that bisects the angle made at that vertex by corresponding sides.

Answer 2

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Answer:

1. Translate P to E; rotate ∆PQR about E until Q is coincident with F; reflect ∆PQR across EF
2. Reflect ∆PQR across line PR; translate R to G; rotate ∆PQR about G until P is coincident with E

Explanation:

The orientations of the triangles are opposite, so a reflection is involved. The various segments are not at right angles to each other, so a rotation other than some multiple of 90° is involved. A translation is needed in order to align the vertices on top of one another.

The rotation is more easily defined if one of the ∆PQR vertices is already on top of its corresponding ∆EFG vertex, so that translation should precede the rotation. The reflection can come anywhere in the sequence.

__

Additional comment

The mapping can be done in two transformations: translate a ∆PQR vertex to its corresponding ∆EFG point; reflect across the line that bisects the angle made at that vertex by corresponding sides.