To assess whether triangles \( \triangle ABC \) and \( \triangle XYZ \) are congruent and to determine what transformations were applied to \( \triangle ABC \) to produce \( \triangle XYZ \), we start by analyzing the coordinates of the points for both triangles:
-
\( \triangle ABC \):
- \( A(-5, 2) \)
- \( B(-3, 2) \)
- \( C(-3, 2) \)
-
\( \triangle XYZ \):
- \( X(-6, 0) \)
- \( Y(-4, 0) \)
- \( Z(-4, -3) \)
Step 1: Verify the Vertices
First, let's verify the coordinates for any discrepancies. Notice that points \( B \) and \( C \) have the same coordinates \( (-3, 2) \), which means \( \triangle ABC \) is technically a degenerate triangle (it collapses into a line segment because two of its points are the same). Due to this, we focus mainly on \( \triangle XYZ \).
Step 2: Analyze the Transformation Between Triangles
To understand what transformation occurred, we observe the movements of the points from \( \triangle ABC \) to \( \triangle XYZ \):
- Translation:
-
Point A (-5, 2) to point X (-6, 0): move left 1 unit and down 2 units. This can be represented as: \[ T_{1}(-5, 2) = (-5 - 1, 2 - 2) = (-6, 0) \]
-
Point B (-3, 2) to point Y (-4, 0): move left 1 unit and down 2 units. Hence, \[ T_{2}(-3, 2) = (-3 - 1, 2 - 2) = (-4, 0) \]
-
Point C (-3, 2) to point Z (-4, -3): move left 1 unit and down 5 units: \[ T_{3}(-3, 2) = (-3 - 1, 2 - 5) = (-4, -3) \]
-
Step 3: Dealing with Degenerate Triangle
Since \( \triangle ABC \) is a degenerate triangle with points lying on a horizontal line segment, and after transformation, \( \triangle XYZ \) has non-collinear points, it indicates a significant difference in structure.
Step 4: Check for Congruence
To determine congruence, we need to check if the triangles have the same shape and size.
- Calculate Lengths of Sides:
- Since \( \triangle ABC \) is degenerate, we can state it does not have a valid area.
- For \( \triangle XYZ\):
- Side \( XY \): \[ XY = \sqrt{((-6) - (-4))^2 + (0 - 0)^2} = \sqrt{(-2)^2} = 2 \]
- Side \( YZ \): \[ YZ = \sqrt{((-4) - (-4))^2 + (0 - (-3))^2} = \sqrt{(0)^2 + (3)^2} = 3 \]
- Side \( ZX \): \[ ZX = \sqrt{((-4) - (-6))^2 + ((-3) - 0)^2} = \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \]
Conclusion
Since \( \triangle ABC \) is degenerate and essentially has no valid area or shape, while \( \triangle XYZ \) is a valid triangle with sides calculated, the two triangles cannot be congruent.
In summary, the transformation observed was a translation, but the two triangles \( \triangle ABC \) and \( \triangle XYZ \) are not congruent due to the degenerate nature of \( \triangle ABC \).