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?

Answer 1

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:

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

1. (-2, 1) → (-2, -1)
2. (0, 1) → (0, -1)
3. (0, 3) → (0, -3)
4. (-1, 4) → (-1, -4)
5. (-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):

1. (-2, 1) → (2, -1)
2. (0, 1) → (0, -1)
3. (0, 3) → (0, -3)
4. (-1, 4) → (1, -4)
5. (-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.