To explore the transformations that were made to triangle ABC to produce triangle XYZ and to assess whether the two triangles are congruent, let's first analyze the coordinates of the points in each triangle.
Triangle ABC:
- A(2, -5)
- B(2, -3)
- C(5, -3)
Triangle XYZ:
- X(0, -6)
- Y(0, -4)
- Z(0, -3)
Step 1: Transformations
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Translation:
- To find how triangle ABC could be transformed to triangle XYZ, we look at the movement of points from one triangle to the other.
- We can compare the x-coordinates first:
- A(2, -5) to X(0, -6): The x-coordinate changes from 2 to 0 (translated 2 units left), and the y-coordinate goes from -5 to -6 (translated 1 unit down).
- B(2, -3) to Y(0, -4): The x-coordinate also changes from 2 to 0 (translated 2 units left), and the y-coordinate changes from -3 to -4 (translated 1 unit down).
- C(5, -3) to Z(0, -3): The x-coordinate changes from 5 to 0 (translated 5 units left), while the y-coordinate remains the same (no vertical translation).
We see there's not a consistent single vector translation that applies to all points since the distances and directions differ. However, we do note that all points of ABC have been shifted to the left.
-
Reflection:
- The movement toward the y-axis (x=0) indicates that there might also be a reflection involved in the transformation, given how the points align vertically after translation.
Step 2: Assessing Congruence
- Distance between Points: To determine if triangles ABC and XYZ are congruent, we need to check if their sides are equal.
-
Calculate the lengths of the sides of triangle ABC:
- AB = \( \sqrt{(2-2)^2 + (-5 - (-3))^2} = \sqrt{0 + (-2)^2} = \sqrt{4} = 2 \)
- BC = \( \sqrt{(5-2)^2 + (-3 - (-3))^2} = \sqrt{3^2} = 3 \)
- AC = \( \sqrt{(5-2)^2 + (-5 - (-3))^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \)
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Now calculate the lengths of the sides of triangle XYZ:
- XY = \( \sqrt{(0-0)^2 + (-6 - (-4))^2} = \sqrt{0 + (-2)^2} = 2 \)
- YZ = \( \sqrt{(0-0)^2 + (-4 - (-3))^2} = \sqrt{0 + (-1)^2} = 1 \)
- XZ = \( \sqrt{(0-0)^2 + (-6 - (-3))^2} = \sqrt{0 + (-3)^2} = 3 \)
-
Conclusion:
- The lengths of the corresponding sides of triangles ABC and XYZ are not equal (AB and XY are equal, but BC and YZ differ), indicating that the triangles are not congruent.
Final Note:
Triangles are not congruent due to the differences in the lengths of sides, and thus the conditions are not satisfied for congruence (SSS congruence postulate). Each transformation from triangle A to triangle X does not yield triangles that have the same dimensions and shapes.