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Use the image to answer the question.

Two triangles, upper A upper B upper C and Upper X upper Y upper Z, are graphed on a coordinate plane. The x-axis ranges from negative 8 to 4 in increments of 1. The y-axis ranges from negative 8 to 6 in increments of 1.

What transformations were made to △ABC
to produce △XYZ
? Assess whether the two triangles are congruent. How do you know? Show your work.

(2 points)

1 answer

To determine the transformations made to triangle \( \triangle ABC \) to produce triangle \( \triangle XYZ \), we need to analyze the coordinates of the vertices of both triangles. Here's the step-by-step process to identify the transformations and assess congruence:

  1. Vertices Identification: Identify the coordinates of the vertices for both triangles:

    • Let the coordinates of \( \triangle ABC \) be \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \).
    • Let the coordinates of \( \triangle XYZ \) be \( X(x_4, y_4) \), \( Y(x_5, y_5) \), and \( Z(x_6, y_6) \).

  2. Determine Transformations:

    • Translation: Check if the coordinates of \( XYZ \) can be derived by translating \( ABC \) (adding or subtracting a constant from the x-coordinates and/or y-coordinates).
    • Rotation or Reflection: If the triangles are oriented differently (e.g., one is flipped or rotated), then we need to assess if a rotation or reflection is involved.
    • Scaling: Determine if the size of the triangles differs.

  3. Check for Congruence:

    • Side Lengths: Calculate the lengths of the sides of both triangles using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
    • Since congruent triangles have equal side lengths, we can compare the side lengths of triangles \( ABC \) and \( XYZ \).
    • Angle Measures: If the side lengths are equal, we can conclude that the triangles are congruent by the Side-Side-Side (SSS) congruence theorem.

  4. Final Assessment: After conducting the above checks:

    • If the side lengths and/or angles are congruent, we conclude that \( \triangle ABC \) is congruent to \( \triangle XYZ \).
    • If transformations were only translations, rotations, or reflections without changing dimensions, the triangles are congruent.

Unfortunately, without the exact coordinates provided in the image, I can't do the calculations directly. However, follow these steps using the coordinates from the image to analyze the transformations and assess congruence. If you can provide the coordinates or specific information on their positions, I can assist you further!