(2, 3) (4, 3) (4, 5) (3, 6) (2, 5). The figure (a) is moved 4 points to the left and 2 blocks down. Specifically, what formal transformation occurred here? Is the new figure (b), after this translation, congruent to the original(a)? If another figure (c), perfectly identical to figure(b), is a reflection in chapter 2 over the x axis, what are the points on the graph? Is figure c congruent to figure b? If Figure B is rotated the house 180 degrees counter clockwise, what are the points of the new figure (d) in Chapter 2 after a counter clockwise rotation of 180 degrees about the origin?After this translation, is figure d congruent to figure c?

Question

(2, 3)  (4, 3) (4, 5) (3, 6) (2, 5). The figure (a) is moved 4 points to the left and 2 blocks down. Specifically, what formal transformation occurred here? Is the new figure (b), after this translation, congruent to the original(a)? If another figure (c), perfectly identical to figure(b), is a reflection in chapter 2 over the x axis, what are the points on the graph? Is figure c congruent to figure b? If Figure B is rotated the house 180 degrees counter clockwise, what are the points of the new figure (d) in Chapter 2 after a counter clockwise rotation of 180 degrees about the origin?After this translation, is figure d congruent to figure c?

GPT-4o mini · Answer

Let's go through each part of the problem step by step.

Step 1: Translation of Figure (a)

The original points of figure (a) are:

(2, 3)
(4, 3)
(4, 5)
(3, 6)
(2, 5)

Transformation: The figure is moved 4 points to the left and 2 blocks down.

To move a point (x, y) left 4 points and down 2 points, we subtract 4 from the x-coordinate and 2 from the y-coordinate: \[ (x, y) \to (x - 4, y - 2) \]

Applying this to each point:

(2, 3) → (2 - 4, 3 - 2) = (-2, 1)
(4, 3) → (4 - 4, 3 - 2) = (0, 1)
(4, 5) → (4 - 4, 5 - 2) = (0, 3)
(3, 6) → (3 - 4, 6 - 2) = (-1, 4)
(2, 5) → (2 - 4, 5 - 2) = (-2, 3)

Thus, the new coordinates for figure (b) are:

(-2, 1)
(0, 1)
(0, 3)
(-1, 4)
(-2, 3)

Step 2: Congruence of Figures (a) and (b)

A translation is a rigid motion which preserves distances and angles. Therefore, figure (b) is congruent to figure (a).

Step 3: Reflection over the x-axis to form figure (c)

To find the points of figure (c) after reflecting figure (b) over the x-axis, we change the sign of the y-coordinate for each point: \[ (x, y) \to (x, -y) \]

Applying this transformation to each point of figure (b):

(-2, 1) → (-2, -1)
(0, 1) → (0, -1)
(0, 3) → (0, -3)
(-1, 4) → (-1, -4)
(-2, 3) → (-2, -3)

Thus, the coordinates for figure (c) are:

(-2, -1)
(0, -1)
(0, -3)
(-1, -4)
(-2, -3)

Step 4: Congruence of Figures (b) and (c)

Reflection is also a rigid motion that preserves distances and angles. Therefore, figure (c) is congruent to figure (b).

Step 5: 180-degree Counterclockwise Rotation of Figure (b) to form Figure (d)

To rotate a point (x, y) 180 degrees counterclockwise around the origin, we use the following transformation: \[ (x, y) \to (-x, -y) \]

Applying this transformation to each point of figure (b):

(-2, 1) → (2, -1)
(0, 1) → (0, -1)
(0, 3) → (0, -3)
(-1, 4) → (1, -4)
(-2, 3) → (2, -3)

Thus, the coordinates for figure (d) are:

(2, -1)
(0, -1)
(0, -3)
(1, -4)
(2, -3)

Step 6: Congruence of Figures (c) and (d)

A 180-degree rotation is also a rigid motion, so figure (d) is congruent to figure (c).

Summary of Results

Figure (b) is congruent to figure (a) (translation)
Figure (c) is congruent to figure (b) (reflection)
Figure (d) is congruent to figure (c) (rotation)

Thus, all figures (a), (b), (c), and (d) are congruent to each other.