Question

Determine the congruence transformation used on △ABC that resulted in △DEF. (1 point) Responses △ABC was shifted 2 units to the right and 2 units up, then rotated 180° (counterclockwise). triangle upper A upper B upper C was shifted 2 units to the right and 2 units up, then rotated 180 degrees (counterclockwise). △ABC was reflected across the y-axis, then shifted 2 units to the right and 2 units up. triangle upper A upper B upper C was reflected across the y -axis, then shifted 2 units to the right and 2 units up. △ABC was shifted 2 units to the left and 2 units up, then reflected across the y-axis. triangle upper A upper B upper C was shifted 2 units to the left and 2 units up, then reflected across the y -axis. △ABC was rotated −180° (clockwise) or 180° (counterclockwise), then shifted 2 units to the right and 2 units up.

Answer 1

To determine the congruence transformation that resulted in triangle \( \triangle DEF \) from triangle \( \triangle ABC \), we will analyze the options provided.

1. Translation followed by Rotation:

   • Shifting \( \triangle ABC \) 2 units to the right and 2 units up, followed by a rotation of 180° counterclockwise could perfectly change its location and orientation.

2. Reflection followed by Translation:

   • Reflecting \( \triangle ABC \) across the y-axis, then shifting it also matches transformations that can lead to a congruent triangle.

3. Translation to the Left and Reflection:

   • Shifting left and then reflecting across the y-axis is another sequence of transformations.

4. Rotation followed by Translation:

   • Rotating \( \triangle ABC \) by 180° and then shifting also produces another combination to get \( \triangle DEF \).

Analysis on outcomes:

• The possibility of a rotation or reflection would keep the size and shape the same, and a translation simply moves the shape without changing it.

To finalize, without specific coordinate information about points \( A, B, C, D, E, F \), it’s hard to directly calculate which transformation applies. However, since translating followed by rotation is a common combination for congruence transformations and includes an orientation change (rotation by 180°), it’s likely the most fitting if we want an ending triangle in a different location yet match in geometry.

Thus, the answer likely is: \( \triangle ABC \) was shifted 2 units to the right and 2 units up, then rotated 180° (counterclockwise).

Answer 2

A (-1,3), B (-2,-1), C (1,0)
D(3,-1), E (4,3), F (1,2)

Answer 3

To determine which congruence transformation was applied to triangle \( \triangle ABC \) with vertices \( A(-1, 3) \), \( B(-2, -1) \), and \( C(1, 0) \) to obtain triangle \( \triangle DEF \) with vertices \( D(3, -1) \), \( E(4, 3) \), and \( F(1, 2) \), we should follow these steps:

1. Calculate Properties of Triangle \( ABC \):

   • Calculate the lengths of the sides using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \):
     • \( AB = \sqrt{((-2) - (-1))^2 + ((-1) - 3)^2} = \sqrt{(-1)^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17} \)
     • \( BC = \sqrt{(1 - (-2))^2 + (0 - (-1))^2} = \sqrt{(3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10} \)
     • \( AC = \sqrt{(1 - (-1))^2 + (0 - 3)^2} = \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \)

2. Calculate Properties of Triangle \( DEF \):

   • \( DE = \sqrt{(4 - 3)^2 + (3 - (-1))^2} = \sqrt{(1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17} \)
   • \( EF = \sqrt{(1 - 4)^2 + (2 - 3)^2} = \sqrt{(-3)^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \)
   • \( DF = \sqrt{(1 - 3)^2 + (2 - (-1))^2} = \sqrt{(-2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \)

3. Verify the Lengths:

   • The sides of both triangles have the same lengths:
     • \( AB \) and \( DE \) both measure \( \sqrt{17} \).
     • \( BC \) and \( EF \) both measure \( \sqrt{10} \).
     • \( AC \) and \( DF \) both measure \( \sqrt{13} \).

4. Check for congruence transformation:

   • The lengths confirm that the triangles are congruent. Now we check the transformation.
   • To determine the translation vector, we can look at one point:
     • Move point \( A(-1, 3) \) to point \( D(3, -1) \):
       • This indicates a translation \( T(x, y) \) where the vector is calculated as:
         • \( T(-1, 3) \rightarrow (3, -1) \) means:
           • \( x: 3 - (-1) = 4 \) (shift right)
           • \( y: -1 - 3 = -4 \) (shift down)
       • Thus, the coordinates are translated by \( T(4, -4) \).

5. Verify Orientation and Check Possible Transforms:

   • After transformation \( ABC \) to \( D \):
     • Check if we can reach \( E \) and \( F \) through subsequent transformations.
     • Since \( D(3, -1) \) is the translated version of \( A(-1, 3) \), we might investigate a rotation afterward if necessary.
     • \( DEF \) has a different arrangement — indicating potential rotational changes within the transformation.

6. Conclusion on Congruence Transformation:

   • From examination of both shapes, they can be obtained by a combination of translation (4 units right and 4 units down) followed by a rotation to achieve orientation alignment.

This corresponds best to "Triangle \( ABC \) was shifted 4 units to the right and 4 units down, then rotated to align."

Without reflecting through points for verifying angles or specific orientation flipped (not part of data analysis), it stands to remain congruently transformed either directly and will likely need further exact reflection verification based on the data from coordinate systems based on points checks.