Math Problem Statement

M is the midpoint of AD.

Triangles A B M and D C M are connected at point M. Sides A B and C D are congruent. The lengths of sides B M and M C are 27.

What single transformation is required to map one of these congruent triangles onto the other?

A. Reflection B. Rotation C. Translation D. Dilation

M is the midpoint of AD. Triangles A B M and D C M are connected at point M. Sides A B and C D are congruent. The lengths of sides B M and M C are 27. What single transformation is required to map one of these congruent triangles onto the other? A. Reflection B. Rotation C. Translation D. Dilation

Solution

In this problem, we are given two triangles ABM\triangle ABM and DCM\triangle DCM, which share point MM as their vertex. We know that the sides ABAB and CDCD are congruent, and BM=MC=27BM = MC = 27. The problem asks us to determine which transformation will map one triangle onto the other.

Given the image and information:

  • The triangles are congruent by the Side-Side-Side (SSS) congruence criterion: AB=CDAB = CD, BM=MCBM = MC, and both triangles share side AM=DMAM = DM.
  • Since these triangles share a vertex at MM, and appear symmetric about a vertical line passing through MM, the appropriate transformation would be a reflection.

The correct transformation to map one triangle onto the other is:

A. Reflection

Would you like further details or explanations about reflections or congruence?

Here are five related questions to deepen your understanding:

  1. How can you identify a line of reflection between two congruent triangles?
  2. What are other types of transformations besides reflection, and how do they differ?
  3. Can two triangles still be congruent if their angles differ but their sides are the same?
  4. What conditions must be met for triangles to be congruent under the SSS criterion?
  5. How can you verify if two figures are congruent using transformations?

Tip: When solving geometric problems involving congruence, always check for symmetry, which often suggests reflection as a transformation!

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Math Problem Analysis

Mathematical Concepts

Geometry
Congruence
Triangle Transformations

Formulas

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Theorems

SSS Congruence Criterion
Reflection Theorem

Suitable Grade Level

Grades 8-10