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

Question

Description of the graph:
Triangle 1: C is located at (-3, 5). A is located at (-5, 2) and B is located at (-3, 2).
Triangle 2: Z is located at (-3, -4). Y is located at (0, -4). And X is located at (0, -6).

Please and thank you for helping me!! :D :)

The graph:

Answer 1

To determine the transformations made to triangle \( \triangle ABC \) to produce triangle \( \triangle XYZ \), we first need to analyze the coordinates of the vertices of both triangles:

Coordinates of Triangle \( ABC \):

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

Coordinates of Triangle \( XYZ \):

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

Step 1: Identify the transformations

1. Translation: We need to find how to translate triangle \( ABC \) to obtain triangle \( XYZ \).
   • The x-coordinates of \( A, B, C \) transform to the x-coordinates of \( X, Y, Z \).
   • The y-coordinates of \( A, B, C \) transform to the y-coordinates of \( X, Y, Z \).

Calculate translation vector

To translate \( \triangle ABC \) to \( \triangle XYZ \):

• The translation needed to move point \( C (-3, 5) \) to point \( Z (-3, -4) \):
  • x-coordinate: No change (from -3 to -3, translation in x is 0).
  • y-coordinate: Move from 5 to -4, which is down by 9 units (5 - 9 = -4).

Thus, the translation vector is:

• Translation Vector = \( (0, -9) \)

Step 2: Apply the translation to the points

Now let's apply the translation \( (0, -9) \) to \( \triangle ABC \):

• New coordinates of \( A \): \[ A' = (-5 + 0, 2 - 9) = (-5, -7) \]
• New coordinates of \( B \): \[ B' = (-3 + 0, 2 - 9) = (-3, -7) \]
• New coordinates of \( C \): \[ C' = (-3 + 0, 5 - 9) = (-3, -4) \]

Points after translation:

• Translated \( A' (-5, -7) \)
• Translated \( B' (-3, -7) \)
• Translated \( C' (-3, -4) \)

Step 3: Compare the triangles

Now we need to examine if \( \triangle ABC \) and \( \triangle XYZ \) are congruent. For triangles to be congruent, their corresponding sides must be equal in length.

Calculate side lengths of triangle \( ABC \):

• Length \( AB \): \[ AB = \sqrt{((-3) - (-5))^2 + ((2) - (2))^2} = \sqrt{(2)^2 + (0)^2} = 2 \]
• Length \( BC \): \[ BC = \sqrt{((-3) - (-3))^2 + ((5) - (2))^2} = \sqrt{(0)^2 + (3)^2} = 3 \]
• Length \( AC \): \[ AC = \sqrt{((-3) - (-5))^2 + ((5) - (2))^2} = \sqrt{(2)^2 + (3)^2} = \sqrt{13} \]

Calculate side lengths of triangle \( XYZ \):

• Length \( XY \): \[ XY = \sqrt{((0) - (0))^2 + ((-4) - (-6))^2} = \sqrt{(0)^2 + (2)^2} = 2 \]
• Length \( YZ \): \[ YZ = \sqrt{((-3) - (0))^2 + ((-4) - (-4))^2} = \sqrt{(-3)^2 + (0)^2} = 3 \]
• Length \( XZ \): \[ XZ = \sqrt{((-3) - (0))^2 + ((-4) - (-6))^2} = \sqrt{(-3)^2 + (2)^2} = \sqrt{13} \]

Step 4: Check congruence

Comparing side lengths of \( \triangle ABC \) and \( \triangle XYZ \):

• \( AB = XY \) (2 = 2)
• \( BC = YZ \) (3 = 3)
• \( AC = XZ \) (\( \sqrt{13} = \sqrt{13} \))

Conclusion

Since the corresponding sides of both triangles are equal in length, we conclude:

• The triangles \( \triangle ABC \) and \( \triangle XYZ \) are congruent by the Side-Side-Side (SSS) congruence criterion.

Final Answer

The transformation was a translation of \( (0, -9) \), and yes, \( \triangle ABC \) is congruent to \( \triangle XYZ \).

Answer 2

could you repeat that without explaining the steps, just showing the work and showing the overall explanation of how you got to the answer and you assessing whether the two triangles are congruent or not? Thank you!

Answer 3

Given Coordinates

Triangle \( ABC \):

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

Triangle \( XYZ \):

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

Transformation Calculation

Translation of Triangle \( ABC \): To find the translation vector, we move from \( C (-3, 5) \) to \( Z (-3, -4) \):

• Translation Vector = \( (0, -9) \)

Apply the Translation

Transpose the points of \( A \), \( B \), and \( C \) using the translation vector:

• \( A' = (-5, 2) + (0, -9) = (-5, -7) \)
• \( B' = (-3, 2) + (0, -9) = (-3, -7) \)
• \( C' = (-3, 5) + (0, -9) = (-3, -4) \)

New Coordinates After Translation

• \( A' (-5, -7) \)
• \( B' (-3, -7) \)
• \( C' (-3, -4) \)

Side Lengths Calculation

1. Triangle \( ABC \):

   • \( AB = \sqrt{((-3) - (-5))^2 + ((2) - (2))^2} = 2 \)
   • \( BC = \sqrt{((-3) - (-3))^2 + ((5) - (2))^2} = 3 \)
   • \( AC = \sqrt{((-3) - (-5))^2 + ((5) - (2))^2} = \sqrt{13} \)

2. Triangle \( XYZ \):

   • \( XY = \sqrt{((0) - (0))^2 + ((-4) - (-6))^2} = 2 \)
   • \( YZ = \sqrt{((-3) - (0))^2 + ((-4) - (-4))^2} = 3 \)
   • \( XZ = \sqrt{((-3) - (0))^2 + ((-4) - (-6))^2} = \sqrt{13} \)

Conclusion

Since:

• \( AB = XY \) (2 = 2)
• \( BC = YZ \) (3 = 3)
• \( AC = XZ \) (\( \sqrt{13} = \sqrt{13} \))

Assessment: The triangles \( \triangle ABC \) and \( \triangle XYZ \) are congruent by the Side-Side-Side (SSS) congruence criterion.