Ask a New Question
Menu

Please work out the following problem on a piece of paper unless you have made arrangements to submit digitally. Then take a picture and upload it below.

A. Describe a sequence of rigid transformations that could be used to show that △ABC ≅△EFD
. Be sure to explain how you know that each pair of corresponding vertices will overlap perfectly.

B. Which Triangle Congruence theorem does this sequence of transformations illustrate? Explain.

1 answer

To solve this problem, we'll break it down into two parts as outlined in A and B.

A. Rigid Transformations to Show △ABC ≅ △EFD

  1. Translation: Start by translating triangle ABC so that vertex A moves to vertex E. This means you slide the entire triangle ABC along a straight path to match position E with A. Since translation moves every point the same distance in the same direction, vertices B and C will move to corresponding points B' and C' such that A' overlaps with E.

  2. Rotation: After translation, the next step is to rotate triangle ABC around point E (the new position of A) until vertex B aligns with vertex F. Since rotations maintain the shape and size of triangles, B will rotate into the position of F.

  3. Reflection (if needed): Finally, if after rotation, triangle ABC does not exactly overlap with triangle EFD, you may need to reflect triangle ABC over a line (like the line through E and F) to ensure that C aligns with D. The reflection will also maintain congruence as it is a rigid transformation.

Explanation of Corresponding Vertices:

  • After the translation, A will coincide with E.
  • After the rotation, B will coincide with F.
  • If a reflection is performed afterwards, C will then coincide with D.

This ensures that △ABC overlaps perfectly with △EFD, confirming congruency.

B. Triangle Congruence Theorem Illustrated

This sequence of transformations illustrates the Rigid Motion Theorem for Congruence. Specifically, it aligns with the Side-Angle-Side (SAS) Congruence Postulate in congruence theorems, as we are ensuring that two sides and the included angle (formed by the moved triangle) remain congruent through the transformations.

  • Translation retains the lengths of sides.
  • Rotation retains the angles (including the included angle).
  • Reflection retains congruence as it does not alter the dimensions or angles of the triangle.

Together, these rigid transformations demonstrate that if one triangle can be transformed into another using a sequence of translations, rotations, and reflections, then the two triangles are congruent, substantiating that △ABC is congruent to △EFD by the SAS Postulate.