which statements correctly describe the transformation in the graph? select all that apply
☐ bcde is rotated 180° counterclockwise to form jklm
☐ bcde is rotated 180° clockwise to form pqrs
☐ jklm is rotated 180° clockwise to form bcde
☐ jklm is rotated 180° counterclockwise to form bcde

Question

Accepted Answer

# Brief Explanations:
1. For "BCDE is rotated 180° counterclockwise to form JKLM": A 180° counterclockwise rotation of a point $(x,y)$ is $(-x,-y)$. Let's check a point from BCDE, say B(2,1). Rotating 180° counterclockwise gives (-2,-1), which is a point in JKLM (e.g., K(-2,-1)? Wait, JKLM's points: Let's assume BCDE has points B(2,1), C(2,2), D(4,2), E(4,1). Rotating 180° counterclockwise: (x,y)→(-x,-y). So B(2,1)→(-2,-1), C(2,2)→(-2,-2), D(4,2)→(-4,-2), E(4,1)→(-4,-1). Which matches JKLM (if J is (-2,-1), K(-2,-2)? Wait maybe labels are different, but the key is 180° rotation (clockwise or counterclockwise) is the same, as 180° rotation in either direction maps (x,y) to (-x,-y).

2. "BCDE is rotated 180° clockwise to form PQRS": PQRS is the green rectangle, say PQRS has points P(-2,1), Q(-4,1), R(-4,2), S(-2,2)? Wait no, original green is PQRS (let's say P(-2,1), Q(-4,1), R(-4,2), S(-2,2)). Rotating BCDE (B(2,1), C(2,2), D(4,2), E(4,1)) 180° clockwise: (x,y)→(-x,-y)? No, 180° clockwise is same as 180° counterclockwise, (x,y)→(-x,-y). Wait BCDE rotated 180° clockwise: B(2,1)→(-2,-1)? No, PQRS is in the second quadrant, positive y. Wait maybe I got labels wrong. Alternatively, PQRS is the green rectangle at (-4,2) to (-2,1)? Wait no, the green rectangle is at x from -4 to -2, y from 1 to 2. So PQRS has points P(-2,1), Q(-4,1), R(-4,2), S(-2,2). BCDE is at x from 2 to 4, y from 1 to 2. Rotating BCDE 180° clockwise: (x,y)→(-x,-y)? No, 180° rotation (clockwise or counter) is (x,y)→(-x,-y). But PQRS is in (-4,-2)? No, green is positive y. Wait maybe the green rectangle is PQRS with y=1 to 2, x=-4 to -2. So BCDE (x=2-4, y=1-2) rotated 180° clockwise: (2,1)→(-2,-1)? No, that's negative y. But PQRS is positive y. So maybe this statement is wrong? Wait no, maybe I mixed up. Wait the blue rectangle is JKLM (negative y), green is PQRS (positive y, left), red is BCDE (positive y, right). So BCDE (red, right) rotated 180° clockwise: (x,y)→(-x,-y)? No, 180° clockwise rotation: the formula is same as counterclockwise for 180°, because 180° is half a circle. So (x,y)→(-x,-y). But PQRS is (x negative, y positive), so BCDE (x positive, y positive) rotated 180° would be (x negative, y negative), which is JKLM (blue). So "BCDE is rotated 180° clockwise to form PQRS" is wrong, because PQRS is positive y. Wait maybe the green rectangle is PQRS with y=1-2, x=-4 to -2 (so (x,y) where x is -4 to -2, y=1-2). So BCDE is (2-4, 1-2). Rotating BCDE 180° clockwise: (x,y)→(-x,-y)? No, 180° clockwise rotation: the direction (clockwise or counter) for 180° doesn't matter, the result is (x,y)→(-x,-y). So BCDE (2,1)→(-2,-1) (blue, JKLM), BCDE (2,1)→(-2,1) would be reflection over y-axis, not 180° rotation. Wait I think I messed up the rotation. Let's recall: 180° rotation (clockwise or counterclockwise) about the origin: (x, y) → (-x, -y). So if BCDE is (2,1), (2,2), (4,2), (4,1), then rotating 180° gives (-2,-1), (-2,-2), (-4,-2), (-4,-1) (JKLM, blue). Rotating BCDE 180° clockwise (same as counter) to get PQRS (green, which is (-2,1), (-4,1), (-4,2), (-2,2)): no, because (-2,1) is not (-x,-y) of (2,1) (which is (-2,-1)). So "BCDE is rotated 180° clockwise to form PQRS" is wrong.

3. "JKLM is rotated 180° clockwise to form BCDE": JKLM is the blue rectangle (negative y), rotating 180° clockwise: (x,y)→(-x,-y)? Wait no, JKLM's points: let's say J(-2,-1), K(-2,-2), L(-4,-2), M(-4,-1). Rotating 180° clockwise: (x,y)→(-x,-y)? No, 180° clockwise rotation: (x,y)→(-x,-y) (same as counter). So J(-2,-1)→(2,1) (which is B), K(-2,-2)→(2,2) (C), L(-4,-2)→(4,2) (D), M(-4,-1)→(4,1) (E). Which is BCDE. So this is correct.

4. "JKLM is rotated 180° counterclockwise to form BCDE": Since 180° counterclockwise rotation is same as 180° clockwise (both map (x,y) to (-x,-y)), so rotating JKLM 180° counterclockwise also maps (x,y)→(-x,-y). So J(-2,-1)→(2,1)=B, etc. So this is also correct, because 180° rotation in either direction is the same. Wait, but 180° clockwise and counterclockwise rotations are identical, as they both result in a half-turn. So:

- "BCDE is rotated 180° counterclockwise to form JKLM": Correct, because (x,y)→(-x,-y) maps BCDE to JKLM.

- "BCDE is rotated 180° clockwise to form PQRS": Incorrect, because PQRS is not (-x,-y) of BCDE.

- "JKLM is rotated 180° clockwise to form BCDE": Correct, because rotating JKLM (blue) 180° clockwise (maps (-x,-y) to (x,y)) gives BCDE (red).

- "JKLM is rotated 180° counterclockwise to form BCDE": Correct, because 180° counterclockwise rotation of JKLM (blue) also maps (-x,-y) to (x,y), same as clockwise.

Wait but let's check the options:

Option 1: BCDE rotated 180° counterclockwise to form JKLM: Correct, as (x,y)→(-x,-y) maps BCDE to JKLM.

Option 2: BCDE rotated 180° clockwise to form PQRS: Incorrect, PQRS is not (-x,-y) of BCDE.

Option 3: JKLM rotated 180° clockwise to form BCDE: JKLM is (-x,-y) of BCDE, so rotating JKLM 180° clockwise (which is (x,y)→(-x,-y)) would take JKLM's (-x,-y) to (x,y), which is BCDE. Correct.

Option 4: JKLM is rotated 180° counterclockwise to form BCDE: Same as above, 180° counterclockwise rotation of JKLM ( (-x,-y) ) is (x,y), which is BCDE. Correct.

Wait but maybe the labels are different. Alternatively, 180° rotation (clockwise or counter) is the same, so rotating a figure 180° clockwise or counterclockwise gives the same result. So:

- BCDE rotated 180° (either direction) gives JKLM (blue).

- JKLM rotated 180° (either direction) gives BCDE (red).

- BCDE rotated 180° to PQRS (green): PQRS is in the second quadrant (x negative, y positive), while BCDE is (x positive, y positive). Rotating BCDE 180° would give (x negative, y negative), not positive y. So PQRS is a reflection over y-axis, not 180° rotation. So "BCDE is rotated 180° clockwise to form PQRS" is wrong.

So correct statements:

- BCDE is rotated 180° counterclockwise to form JKLM (correct, as 180° rotation maps to JKLM).

- JKLM is rotated 180° clockwise to form BCDE (correct, as 180° rotation maps back).

- JKLM is rotated 180° counterclockwise to form BCDE (correct, same as clockwise for 180°).

Wait but maybe the first option: "BCDE is rotated 180° counterclockwise to form JKLM" – let's check coordinates. Suppose BCDE has vertices at B(2,1), C(2,2), D(4,2), E(4,1). Rotating 180° counterclockwise: (x,y) → (-x,-y). So B(2,1) → (-2,-1), C(2,2) → (-2,-2), D(4,2) → (-4,-2), E(4,1) → (-4,-1). Which are the vertices of JKLM (assuming J is (-2,-1), K(-2,-2), L(-4,-2), M(-4,-1)). So that's correct.

"JKLM is rotated 180° clockwise to form BCDE": JKLM's vertices are (-2,-1), (-2,-2), (-4,-2), (-4,-1). Rotating 180° clockwise: (x,y) → (-x,-y)? No, 180° clockwise rotation is (x,y) → (-x,-y)? Wait no, 180° rotation (clockwise or counter) is (x,y) → (-x,-y). So rotating JKLM's (-2,-1) 180° clockwise: (-2,-1) → (2,1) (which is B), (-2,-2) → (2,2) (C), (-4,-2) → (4,2) (D), (-4,-1) → (4,1) (E). So that's BCDE. Correct.

"JKLM is rotated 180° counterclockwise to form BCDE": Same as above, because 180° counterclockwise rotation is also (x,y) → (-x,-y). So (-2,-1) → (2,1), etc. Correct.

"BCDE is rotated 180° clockwise to form PQRS": PQRS is the green rectangle, say P(-2,1), Q(-4,1), R(-4,2), S(-2,2). Rotating BCDE (2,1) 180° clockwise: (2,1) → (-2,-1), which is not P(-2,1). So this is wrong.

So correct options are:

- BCDE is rotated 180° counterclockwise to form JKLM

- JKLM is rotated 180° clockwise to form BCDE

- JKLM is rotated 180° counterclockwise to form BCDE

Wait but maybe the problem's labels are different. Alternatively, maybe "BCDE is rotated 180° clockwise to form PQRS" is wrong, as PQRS is a different figure. So the correct statements are the first, third, and fourth? Wait no, let's re-express:

1. BCDE rotated 180° counterclockwise → JKLM: Correct.

2. BCDE rotated 180° clockwise → PQRS: Incorrect (PQRS is not (-x,-y) of BCDE).

3. JKLM rotated 180° clockwise → BCDE: Correct (JKLM is (-x,-y) of BCDE, so rotating 180° clockwise (which is (x,y)→(-x,-y)) takes JKLM's (-x,-y) to (x,y), i.e., BCDE).

4. JKLM rotated 180° counterclockwise → BCDE: Correct (same as clockwise for 180° rotation).

So the correct options are the first, third, and fourth? Wait but maybe the problem's PQRS is the green rectangle, which is a reflection over y-axis, not 180° rotation. So the answer is:

- BCDE is rotated 180° counterclockwise to form JKLM (correct)

- JKLM is rotated 180° clockwise to form BCDE (correct)

- JKLM is rotated 180° counterclockwise to form BCDE (correct)

But let's check the original options:

Options:

1. BCDE is rotated 180° counterclockwise to form JKLM – Correct.

2. BCDE is rotated 180° clockwise to form PQRS – Incorrect (PQRS is not 180° rotation of BCDE).

3. JKLM is rotated 180° clockwise to form BCDE – Correct.

4. JKLM is rotated 180° counterclockwise to form BCDE – Correct.

So the correct options are 1, 3, 4? Wait maybe the labels of the rectangles are different. Alternatively, maybe PQRS is the green rectangle, which is the 180° rotation? No, because green is in the second quadrant (x negative, y positive), BCDE is in first quadrant (x positive, y positive). 180° rotation would be third quadrant (x negative, y negative), which is JKLM (blue). So PQRS is a reflection over y-axis, not 180° rotation. So option 2 is wrong.

Thus, the correct statements are:

- BCDE is rotated 180° counterclockwise to form JKLM

- JKLM is rotated 180° clockwise to form BCDE

- JKLM is rotated 180° counterclockwise to form BCDE

# Answer:
- BCDE is rotated 180° counterclockwise to form JKLM
- JKLM is rotated 180° clockwise to form BCDE
- JKLM is rotated 180° counterclockwise to form BCDE

(Note: If the problem's rectangle labels differ slightly, the key is that a 180° rotation (clockwise or counterclockwise) maps $(x,y)$ to $(-x,-y)$, so rotating a figure 180° in either direction to its "opposite" quadrant (e.g., first to third, third to first) is correct, while rotating to a quadrant with the same $y$-sign (first to second) is incorrect.)

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