Which of these triangle pairs can be mapped to each other using a single reflection?

Which of these triangle pairs can be mapped to each other using a single reflection?

December 12, 2024

Question: Which of these triangle pairs can be mapped to each other using a single reflection?

Answer: The pair of triangles shown in the last image (the pair identified as EFG and LMN) can be mapped to each other using a single reflection.

Explanation:

Step1: Identify Congruent Triangles

From the provided images, we are looking for two triangles that are congruent and differ by a mirror-image orientation rather than a rotation or translation.

Step2: Recognize Reflection Characteristics

A reflection maps every point of one figure to a corresponding point on another figure across a line of symmetry. After reflection, the corresponding angles and sides remain equal, but the orientation is reversed (like looking in a mirror).

Step3: Check the Pairs of Triangles

  • In the earlier figures, the arrangement of points and side-angle markings suggests the pairs might be related by more complex transformations.
  • In the final image, the pair of triangles (EFG and LMN) have corresponding sides and angles marked the same. Their orientation suggests that flipping one triangle over an appropriate line would match all corresponding points.

Extended Knowledge:

Reflections in Geometry

Reflections produce mirror images of figures across a line (axis of reflection). The image after a reflection maintains shape and size but reverses orientation.

Distinguishing Reflection from Other Transformations

  • Rotation: Turns a figure around a fixed point.
  • Translation: Slides a figure without changing its orientation.
  • Reflection: Flips a figure over a line, producing a reversed orientation.

By examining orientation and corresponding congruent parts, we can determine that a single reflection is the appropriate transformation for the final pair of triangles.