select all the sequences of transformations that could return a figure to its original position. reflect the figure over a line and then reflect back over the same one. translate the figure 1 unit to the right, then 4 units to the left, and then 3 units to the right. reflect the figure over one line and then reflect over a different line. rotate the figure 90° counterclockwise around a point and then 270° counterclockwise around the same point. rotate the figure 90° counterclockwise around a point and then 270° clockwise around the same point.

Question

Accepted Answer

To solve this, we analyze each option:

### Option 1: Reflect over a line, then reflect back over the same line
Reflecting a figure over a line and then reflecting it back over the same line undoes the first reflection. So the figure returns to its original position. This works.

### Option 2: Translate 1 right, 4 left, 3 right
Let the original position be $ x $. After translation: $ x + 1 - 4 + 3 = x + (1 - 4 + 3) = x + 0 = x $. Wait, no—wait, $ 1 - 4 = -3 $, then $ -3 + 3 = 0 $. Wait, actually: $ x + 1 $ (right 1), then $ x + 1 - 4 = x - 3 $ (left 4), then $ x - 3 + 3 = x $ (right 3). Wait, that sums to 0? Wait, no: $ 1 - 4 + 3 = 0 $, so the total translation is 0. Wait, but let’s check again: 1 right, 4 left (net: 3 left), then 3 right (net: 0). So the figure returns? Wait, no—wait, $ 1 - 4 + 3 = 0 $, so the total displacement is 0. Wait, but let’s test with a point: original at $ (0,0) $. After 1 right: $ (1,0) $. After 4 left: $ (1 - 4, 0) = (-3, 0) $. After 3 right: $ (-3 + 3, 0) = (0, 0) $. Oh, it does return! Wait, but wait—did I miscalculate? Wait, $ 1 - 4 + 3 = 0 $, so the total translation is 0. So this works? Wait, no—wait, the problem says "select all"—but let’s check the other options first.

### Option 3: Reflect over one line, then a different line
Reflecting over two different lines does not necessarily return the figure to its original position (unless the lines are related in a specific way, but generally, this is not guaranteed). So this does not work.

### Option 4: Rotate 90° counterclockwise, then 270° counterclockwise (same point)
Total rotation: $ 90^\circ + 270^\circ = 360^\circ $ counterclockwise. A $ 360^\circ $ rotation returns the figure to its original position. This works.

### Option 5: Rotate 90° counterclockwise, then 270° clockwise (same point)
A $ 270^\circ $ clockwise rotation is equivalent to a $ 90^\circ $ counterclockwise rotation (since $ 360^\circ - 270^\circ = 90^\circ $ counterclockwise). Wait, no: $ 90^\circ $ counterclockwise + $ 270^\circ $ clockwise = $ 90^\circ - 270^\circ = -180^\circ $ (or $ 180^\circ $ clockwise), which does not return to original. Wait, no—wait, $ 270^\circ $ clockwise is the same as $ -270^\circ $ counterclockwise. So total rotation: $ 90^\circ + (-270^\circ) = -180^\circ $ (or $ 180^\circ $ clockwise), which is not $ 360^\circ $. Wait, no—wait, let’s think: rotating 90° CCW, then 270° CW. 270° CW is the same as 90° CCW (because 360 - 270 = 90). Wait, no: 90° CCW, then 270° CW: the total rotation is $ 90^\circ - 270^\circ = -180^\circ $ (180° CW), which does not return to original. Wait, no—wait, maybe I made a mistake. Wait, 90° CCW, then 270° CW: 270° CW is three 90° CW turns. 90° CCW is one 90° CCW turn. So total: 90° CCW + 270° CW = (90° CCW) + (270° CW) = (90° CCW) + (-270° CCW) = -180° CCW (or 180° CW). So the figure is rotated 180°, not back to original. Wait, but wait—no, wait: 90° CCW, then 270° CW. Let’s take a point (1,0). Rotate 90° CCW: (0,1). Rotate 270° CW: (0,1) rotated 270° CW is (1,0) (since 270° CW is (x,y) → (y, -x); (0,1) → (1, 0)). Oh! Wait, that works. Wait, let’s check: (1,0) → 90° CCW: (0,1). Then 270° CW: (0,1) rotated 270° CW: the formula for 270° CW is $ (x,y) \to (y, -x) $. So (0,1) → (1, 0). Which is the original point. Wait, so why? Because 90° CCW + 270° CW = 360° total rotation? Wait, 90 + 270 = 360. Oh! Because 270° CW is the same as -270° CCW, so 90° CCW + (-270° CCW) = -180° CCW? No, that can’t be. Wait, no—rotation direction: CCW is positive, CW is negative. So 90° CCW is +90°, 270° CW is -270°. Total: +90° - 270° = -180° (180° CW). But when we tested with (1,0), we got back to original. Wait, that’s a contradiction. Wait, no—wait, (0,1) rotated 270° CW: let’s do it step by step. 90° CW: (1,0); 180° CW: (0,-1); 270° CW: (-1,0)? Wait, no—wait, rotation matrices: 90° CCW: $ \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} $; 270° CW is the same as 90° CCW (since 270° CW = 360° - 270° = 90° CCW). Wait, no—270° CW is equivalent to 90° CCW? No, 270° CW is three 90° CW turns. 90° CW: $ \begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix} $. So 270° CW is applying the 90° CW matrix three times. Let’s apply it to (0,1):

First 90° CW: $ (0,1) \to (1, 0) $ (matrix: $ \begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix} \begin{pmatrix} 0 \ 1 \end{pmatrix} = \begin{pmatrix} 1 \ 0 \end{pmatrix} $)

Second 90° CW: $ (1,0) \to (0, -1) $

Third 90° CW: $ (0,-1) \to (-1, 0) $

Wait, so (0,1) rotated 270° CW is (-1, 0), not (1,0). So my earlier test was wrong. So (1,0) → 90° CCW: (0,1) → 270° CW: (-1, 0), which is not original. So Option 5 does not work. Wait, but maybe I messed up the rotation direction. Let’s re-express:

- 90° counterclockwise: (x,y) → (-y, x)
- 270° clockwise: (x,y) → (y, -x) (because 270° clockwise is the same as 90° counterclockwise? No, wait: 90° clockwise is (x,y) → (y, -x); 180° clockwise is (x,y) → (-x, -y); 270° clockwise is (x,y) → (-y, x) (which is 90° counterclockwise). Oh! Wait, yes: 270° clockwise is equivalent to 90° counterclockwise. Because 360° - 270° = 90°, so 270° clockwise = 90° counterclockwise. So (x,y) rotated 270° clockwise is the same as (x,y) rotated 90° counterclockwise: (-y, x).

So let’s redo the test: (1,0) rotated 90° CCW: (-0, 1) = (0,1). Then rotated 270° CW (which is 90° CCW): (-1, 0). Wait, that’s not original. So Option 5 does not work. Wait, but earlier I thought 90 + 270 = 360, but direction matters. 90° CCW + 270° CW = 90° CCW - 270° CCW = -180° CCW (180° CW), which is a 180° rotation, not 360°. So Option 5 does not work.

Wait, let’s re-analyze Option 2: Translate 1 right, 4 left, 3 right. The total horizontal displacement is $ 1 - 4 + 3 = 0 $, so the figure returns to original. Wait, but let’s check with a point: (0,0) → (1,0) (1 right) → (1 - 4, 0) = (-3, 0) (4 left) → (-3 + 3, 0) = (0, 0) (3 right). So it does return. So Option 2 works? Wait, but the problem says "select all"—but let’s confirm each option:

1. Reflect over line, then back: works (undoes reflection).
2. Translate 1 right, 4 left, 3 right: total translation 0, works.
3. Reflect over two different lines: does not work (unless lines are perpendicular and intersect at the center, but generally no).
4. Rotate 90° CCW, then 270° CCW: total 360° CCW, works (returns to original).
5. Rotate 90° CCW, then 270° CW: total 180° rotation, does not work.

Wait, but maybe I made a mistake with Option 2. Let’s check the arithmetic: 1 (right) - 4 (left) + 3 (right) = (1 + 3) - 4 = 4 - 4 = 0. So total translation is 0, so the figure returns. So Option 2 works.

Wait, but let’s check the original options again:

- Option 1: Reflect over same line twice: works.
- Option 2: Translate 1 right, 4 left, 3 right: total translation 0, works.
- Option 3: Reflect over two different lines: no.
- Option 4: Rotate 90° CCW + 270° CCW = 360°: works.
- Option 5: Rotate 90° CCW + 270° CW = 180°: no.

But wait, let’s confirm Option 2: the problem says "translate the figure 1 unit to the right, then 4 units to the left, and then 3 units to the right". So the net translation is $ 1 - 4 + 3 = 0 $, so the figure ends up at the original position. So this works.

Wait, but maybe the intended answer is Option 1, Option 4, and Option 5? No, our test with Option 5 showed it doesn’t work. Wait, maybe I messed up the rotation for Option 5. Let’s re-express:

- 90° counterclockwise: (x,y) → (-y, x)
- 270° clockwise: (x,y) → (y, -x) (wait, no—270° clockwise is (x,y) → (-y, x), which is the same as 90° counterclockwise. Wait, no: 90° clockwise is (x,y) → (y, -x); 180° clockwise is (x,y) → (-x, -y); 270° clockwise is (x,y) → (-y, x) (which is 90° counterclockwise). So 90° CCW followed by 270° CW is 90° CCW + 90° CCW = 180° CCW, which is a 180° rotation, not 360°. So Option 5 does not work.

Wait, let’s check Option 4 again: 90° CCW + 270° CCW = 360° CCW, which is a full rotation, so the figure returns. Correct.

Option 1: Reflect over line, then back: the second reflection undoes the first, so returns to original. Correct.

Option 2: Translate 1 right, 4 left, 3 right: net translation 0, so returns. Correct? Wait, but let’s check with a different point: (2,3). 1 right: (3,3). 4 left: (3 - 4, 3) = (-1, 3). 3 right: (-1 + 3, 3) = (2, 3). Yes, it returns. So Option 2 works.

Wait, but maybe the problem considers "translate 1 right, 4 left, 3 right" as net 0, so it works. So the correct options are:

- Reflect the figure over a line and then reflect back over the same one.
- Translate the figure 1 unit to the right, then 4 units to the left, and then 3 units to the right.
- Rotate the figure 90° counterclockwise around a point and then 270° counterclockwise around the same point.

Wait, but let’s check Option 5 again. If we rotate 90° CCW, then 270° CW, is that equivalent to 360°? Wait, 90° CCW is +90°, 270° CW is -270°, so total is -180° (180° CW), which is not 360°. So Option 5 does not work.

So the correct options are:

1. Reflect the figure over a line and then reflect back over the same one.
2. Translate the figure 1 unit to the right, then 4 units to the left, and then 3 units to the right.
4. Rotate the figure 90° counterclockwise around a point and then 270° counterclockwise around the same point.

Wait, but let’s confirm with the rotation in Option 4: 90 + 270 = 360, so a full rotation, which returns the figure. Correct.

So the answer is the first, second, and fourth options.

# Answer:
- Reflect the figure over a line and then reflect back over the same one.
- Translate the figure 1 unit to the right, then 4 units to the left, and then 3 units to the right.
- Rotate the figure 90° counterclockwise around a point and then 270° counterclockwise around the same point.

(In boxed form for each selected option, but since it’s multiple selection, we list the descriptions.)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Option 1: Reflect over a line, then reflect back over the same line

Option 2: Translate 1 right, 4 left, 3 right

Option 3: Reflect over one line, then a different line

Option 4: Rotate 90° counterclockwise, then 270° counterclockwise (same point)

Option 5: Rotate 90° counterclockwise, then 270° clockwise (same point)

Answer: