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Use the image to answer the question.Two triangles, upper A upper is (-5,2) B upper is (-3,2) C is (-3,5)and Upper X is (0,-6) upper Y is (0,-4) upper Z is (-3,-4), 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 that were made to triangle ABC to produce triangle XYZ and to assess whether the two triangles are congruent, let's analyze the coordinates of the vertices of both triangles.

The vertices of triangle ABC are:

  • A(-5, 2)
  • B(-3, 2)
  • C(-3, 5)

The vertices of triangle XYZ are:

  • X(0, -6)
  • Y(0, -4)
  • Z(-3, -4)

Step 1: Identify the transformations

First, let's identify the change in the coordinates of each triangle's vertices:

  1. Translation:

    • To find the translation vector from triangle ABC to triangle XYZ, we compare the coordinates of corresponding points:
      • Transform A(-5, 2) → X(0, -6):
        • Change in x: 0 - (-5) = 5 (right)
        • Change in y: -6 - 2 = -8 (down)
      • This suggests a translation vector \( (5, -8) \).

  2. Checking other vertices against the translation vector:

    • B(-3,2) should translate similarly:
      • B(-3, 2) → (0, -4):
        • Change x: 0 - (-3) = 3 (right)
        • Change y: -4 - 2 = -6 (down)
        • This does not match the same translation.
    • C(-3, 5) should translate similarly:
      • C(-3, 5) → (-3, -4):
        • Change x: -3 - (-3) = 0 (no change in x)
        • Change y: -4 - 5 = -9 (down)
        • Again does not match the same translation.

From this, we can hypothesize that with different transformations applied, we might be looking at a composite transformation (translation followed by some other modification).

Step 2: Assessing congruency

To find whether triangles ABC and XYZ are congruent, we need to compare the lengths of their sides:

  1. Finding lengths in triangle ABC:
    • AB: Distance between A(-5, 2) and B(-3, 2):
      • Length = \(|-5 - (-3)| = 2\)
    • BC: Distance between B(-3, 2) and C(-3, 5):
      • Length = \(|2 - 5| = 3\)
    • CA: Distance between C(-3, 5) and A(-5, 2):
      • Length = \(\sqrt{((-3) - (-5))^2 + (5 - 2)^2} = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13}\)

Triangle ABC has side lengths: \( 2, 3, \sqrt{13} \).

  1. Finding lengths in triangle XYZ:
    • XY: Distance between X(0, -6) and Y(0, -4):
      • Length = \(|-6 - (-4)| = 2\)
    • YZ: Distance between Y(0, -4) and Z(-3, -4):
      • Length = \(|0- (-3)| = 3\)
    • ZX: Distance between Z(-3, -4) and X(0, -6):
      • Length = \(\sqrt{((-3) - (0))^2 + (-4 - (-6))^2} = \sqrt{(3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13}\)

Triangle XYZ has side lengths: \( 2, 3, \sqrt{13} \).

Conclusion

Since the side lengths of triangles ABC and XYZ are equal:

  • Triangle ABC: \( 2, 3, \sqrt{13} \)
  • Triangle XYZ: \( 2, 3, \sqrt{13} \)

This indicates that triangles ABC and XYZ are congruent. Thus, the transformations involved a series of translations and possibly reflections or rotations which maintain side length congruency.

Therefore, we conclude that triangles ABC and XYZ are congruent.