a. rotate the figure around a point 90° clockwise and then rotate it another 90° clockwise
b. reflect the figure over one line and then reflect over a different line.
c. translate the figure 3 units down, then 5 units up, and then 2 units down.
d. reflect the figure over one line and then reflect over the same line.
e. rotate the figure 90° counterclockwise around a point and then 270° counterclockwise around the same point.
f. rotate a figure 180° counterclockwise, then reflect it over a vertical line.

Question

Accepted Answer

To determine which transformations result in the original figure (identity transformation), we analyze each option:

### Option A:
- Rotate 90° clockwise, then another 90° clockwise.
  Total rotation: $ 90^\circ + 90^\circ = 180^\circ $ clockwise. This is not the identity (needs $ 360^\circ $ rotation).

### Option B:
- Reflect over one line, then reflect over a **different** line.
  Two reflections over non - parallel lines result in a rotation, not the identity.

### Option C:
- Translate 3 units down, 5 units up, then 2 units down.
  Net translation: $ - 3+5 - 2=0 $ (down is negative, up is positive). So the total translation is 0, meaning the figure returns to its original position. This is the identity transformation.

### Option D:
- Reflect over the same line twice.
  Reflecting a figure over the same line twice is equivalent to the identity transformation (the figure maps back to itself). Wait, but let's re - check. Wait, no—wait, when we reflect a figure over a line, reflecting it again over the same line will reverse the first reflection. So if $ R $ is the reflection transformation, $ R\circ R $ (applying $ R $ twice) is the identity. But wait, let's check the other options again. Wait, in option C, the translations: 3 down, 5 up (which is a net of 2 up), then 2 down. So $ 3$ down $ + 5$ up $= 2$ up, then $ 2$ down: $ 2$ up $+ 2$ down $ = 0$. So translation net is 0. In option D, reflecting over the same line twice: let's recall the property of reflections. If you reflect a point $ P $ over a line $ l $ to get $ P' $, then reflecting $ P' $ over $ l $ gives back $ P $. So reflecting over the same line twice is the identity. But wait, the problem—maybe I made a mistake. Wait, no, let's re - examine the options. Wait, the original question—maybe the user is asking which transformation is equivalent to the identity. Wait, but let's check option E: Rotate 90° counterclockwise, then 270° counterclockwise around the same point. Total rotation: $ 90^\circ+270^\circ = 360^\circ $ counterclockwise, which is the identity (a full rotation). Option F: Rotate 180° counterclockwise, then reflect over a vertical line. A 180° rotation and a reflection are not equivalent to the identity.

Wait, I think I made a mistake earlier. Let's re - analyze each option:

#### Option A:
Rotate 90° clockwise, then another 90° clockwise. Total rotation: $ 90 + 90=180^\circ $ clockwise. Not identity.

#### Option B:
Reflect over one line, then reflect over a different line. Two reflections over intersecting lines result in a rotation (angle twice the angle between the lines). Not identity.

#### Option C:
Translate 3 units down ($ T_{(0, - 3)} $), then 5 units up ($ T_{(0,5)} $), then 2 units down ($ T_{(0, - 2)} $). The composition of translations is $ T_{(0, - 3+5 - 2)}=T_{(0,0)} $, which is the identity translation (no change in position). So this is the identity.

#### Option D:
Reflect over the same line twice. Let $ R_l $ be the reflection over line $ l $. Then $ R_l\circ R_l $ (applying $ R_l $ twice) is the identity, because reflecting a point over $ l $ and then over $ l $ again brings it back to the original position.

#### Option E:
Rotate 90° counterclockwise ($ R_{90^{\circ},ccw} $), then 270° counterclockwise ($ R_{270^{\circ},ccw} $) around the same point. The composition is $ R_{90 + 270=360^{\circ},ccw} $, which is the identity rotation (a full rotation).

#### Option F:
Rotate 180° counterclockwise, then reflect over a vertical line. A 180° rotation and a reflection are not the identity.

But the question—maybe the user is asking which transformation is equivalent to the identity. If we assume that the question is about which transformation leaves the figure unchanged (identity), then:

- Option C: Translations cancel out ($ 3$ down, $ 5$ up, $ 2$ down: net $ 0$ translation).
- Option D: Reflecting over the same line twice is identity.
- Option E: Rotating $ 90 + 270 = 360^\circ $ is identity.

But since the problem is likely about which transformation is the identity, and if we have to choose one (maybe the question is about which transformation is equivalent to doing nothing), let's re - check the translation in option C:

3 units down: $ y\to y - 3$

5 units up: $ y\to y - 3+5=y + 2$

2 units down: $ y\to y + 2-2=y$

So the $ y$ - coordinate (and $ x$ - coordinate, since there's no horizontal translation) returns to the original. So the translation composition is the identity.

For option D: Reflecting over the same line twice: if you have a point $ (x,y) $, reflecting over line $ l $ gives $ (x',y') $, reflecting $ (x',y') $ over $ l $ gives back $ (x,y) $. So that's also identity.

For option E: Rotating 90° counterclockwise and then 270° counterclockwise around the same point: the total rotation is $ 90 + 270=360^\circ $ counterclockwise, which is the same as not rotating (identity rotation).

If we have to choose the most straightforward one, option C is about translations that cancel out, option D is about two reflections over the same line, and option E is about two rotations that add up to 360°.

But maybe the question is which transformation is equivalent to the identity. If we assume the question is from a basic transformation perspective, let's take option C:

The translations: 3 down, 5 up (net 2 up), then 2 down (net 0). So the figure ends up in the same position.

So the answer depends on the exact question, but if we consider the translation composition in option C, it is an identity transformation.

# Brief Explanations:
- **Option A**: Total rotation is $ 180^\circ $, not identity.
- **Option B**: Two reflections over different lines result in a rotation, not identity.
- **Option C**: Translations ($ 3$ down, $ 5$ up, $ 2$ down) net to $ 0 $, so the figure returns to its original position (identity transformation).
- **Option D**: Reflecting over the same line twice is identity, but let's confirm the question's intent.
- **Option E**: Total rotation is $ 360^\circ $ (identity), but translations in C are more direct for "doing nothing" via cancellation.
- **Option F**: Rotation + reflection is not identity.

# Answer:
C. Translate the figure 3 units down, then 5 units up, and then 2 units down.

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