Question

question 1
10 pts
g.gsr.3.2 (lc)
how many degrees did parallelogram
abcd rotate?
image of a coordinate grid with parallelogram abcd and its rotated image abcd
accessible description
270° counter-clockwise
90° counter-clockwise
180°
270° clockwise

Explanation:

Step1: Analyze Rotation Direction and Angle

To determine the rotation angle, we can track a point, say \( A(1,1) \) to \( A'(1,-1) \)? Wait, no, let's check coordinates. Original \( A \) is at \( (1,1) \), \( A' \) is at \( (1,-1) \)? Wait, no, looking at the grid: Original \( A \): (1,1), \( A' \): (1,-1)? Wait, no, the lower figure: \( A' \) is (1,-1)? Wait, no, the original parallelogram ABCD: A(1,1), B(4,1), D(2,3), C(5,3). The rotated one \( A'(1,-1) \), \( B'(1,-4) \), \( D'(3,-2) \), \( C'(3,-5) \)? Wait, maybe better to use the direction. A 90-degree counter-clockwise rotation of a point \( (x,y) \) is \( (-y,x) \). Let's take point A(1,1). 90 CCW: (-1,1)? No, that's not. Wait, 90-degree clockwise: (y,-x). For A(1,1), 90 CW: (1,-1), but \( A' \) is (1,-1)? Wait, no, original A is (1,1), \( A' \) is (1,-1)? Wait, the grid: original A is at (1,1) (x=1, y=1). \( A' \) is at (1,-1)? No, looking at the lower figure, \( A' \) is at (1,-1)? Wait, the y-axis: original A is at y=1, \( A' \) is at y=-1? Wait, no, the lower figure: \( A' \) is at (1,-1)? Wait, maybe I misread. Wait, original ABCD: A(1,1), B(4,1), D(2,3), C(5,3). Rotated \( A'(1,-1) \), \( B'(1,-4) \), \( D'(3,-2) \), \( C'(3,-5) \)? Wait, no, let's check the vectors. Alternatively, the rotation from ABCD to \( A'B'C'D' \): if we look at the direction, a 90-degree clockwise rotation? Wait, no, 90-degree counter-clockwise: let's take vector AB: from A(1,1) to B(4,1), which is (3,0) (right along x-axis). After rotation, vector \( A'B' \): from \( A'(1,-1) \) to \( B'(1,-4) \), which is (0,-3) (down along y-axis). A 90-degree counter-clockwise rotation of (3,0) is (0,3) (up), but here it's (0,-3) (down), which is 90-degree clockwise? Wait, no, 90-degree clockwise rotation of (3,0) is (0,-3), which matches \( A'B' \). Wait, but the options include 90 CCW, 270 CW (which is 90 CCW), etc. Wait, maybe I made a mistake. Let's take point D(2,3). 90-degree counter-clockwise rotation: (-3,2). Is that \( D' \)? \( D' \) is at (3,-2)? No. Wait, 90-degree clockwise rotation of D(2,3) is (3,-2), which is \( D' \) (3,-2). Yes! So D(2,3) rotated 90 degrees clockwise is (3,-2), which is \( D' \). So the rotation is 90 degrees clockwise, which is equivalent to 270 degrees counter-clockwise. But the options: let's check the options. The options are 270 CCW, 90 CCW, 180, 270 CW. Wait, 90-degree clockwise is 270-degree counter-clockwise? No, 90 CW = 270 CCW? Wait, 360 - 90 = 270, so rotating 90 degrees clockwise is the same as 270 degrees counter-clockwise. But let's check the points. Wait, point A(1,1): 90 CW rotation is (1,-1), which is \( A' \) (1,-1). Yes! So A(1,1) → (1,-1) is 90 CW. So the rotation is 90 degrees clockwise, which is option D? Wait, no, the options are:

- 270° Counter-Clockwise

- 90° Counter-Clockwise

- 180°

- 270° Clockwise

Wait, 90-degree clockwise is 270-degree counter-clockwise? No, 90 CW is 270 CCW? Wait, no: 90 CW rotation: (x,y) → (y, -x). 270 CCW rotation: (x,y) → (y, -x) as well. Yes! Because 270 CCW is the same as 90 CW. So both 90 CW and 270 CCW give the same transformation. But let's check the options. The correct answer is 90° Counter-Clockwise? No, wait, no. Wait, let's take point A(1,1). 90 CCW rotation: (-1,1). Is that \( A' \)? No, \( A' \) is (1,-1). So 90 CW: (1,-1), which is \( A' \). So the rotation is 90 degrees clockwise, which is 270 degrees counter-clockwise? Wait, no, 90 CW is 270 CCW? Wait, 360 - 90 = 270, so rotating 90 degrees clockwise is the same as rotating 270 degrees counter-clockwise. But the options have 270 CCW as an option. Wait, but when we rotate 90 degrees clockwise, the direction is clockwise, but 270 CCW is the same as 90 CW. Wait, maybe the answer is 90° Counter-Clockwise? No, that would be (-y,x). For A(1,1), that would be (-1,1), which is not \( A' \). So 90 CW is (y,-x) = (1,-1), which is \( A' \). So the rotation is 90 degrees clockwise, which is 270 degrees counter-clockwise? Wait, no, 90 CW is 270 CCW? Wait, no, 90 CW is 270 CCW? Let's see: a full rotation is 360. 90 CW: 360 - 90 = 270 CCW. So yes, 90 CW = 270 CCW. But the options include 90° Counter-Clockwise, which is different. Wait, maybe I made a mistake. Let's check another point. Point B(4,1). 90 CW rotation: (1,-4). Is \( B' \) at (1,-4)? Yes! So B(4,1) → (1,-4) is (y,-x) = (1,-4). Yes! So B(4,1) rotated 90 CW is (1,-4), which is \( B' \). So that's correct. So the rotation is 90 degrees clockwise, which is 270 degrees counter-clockwise? Wait, no, 90 CW is 90 degrees clockwise, which is option D: 270° Clockwise? No, 270 CW is 90 CCW. Wait, I'm confused. Let's recall the rotation rules:

- 90° Counter-Clockwise (CCW): (x, y) → (-y, x)

- 90° Clockwise (CW): (x, y) → (y, -x)

- 180°: (x, y) → (-x, -y)

- 270° CCW: (x, y) → (y, -x) [same as 90° CW]

- 270° CW: (x, y) → (-y, x) [same as 90° CCW]

Ah! So 270° CCW is the same as 90° CW, and 270° CW is the same as 90° CCW. So for point A(1,1):

- 90° CW: (1, -1) → matches \( A' \) (1, -1)

- 270° CCW: (1, -1) → same as 90° CW

So the rotation is 90° CW, which is equivalent to 270° CCW. But the options:

- 270° Counter-Clockwise: this is 90° CW, which matches.

- 90° Counter-Clockwise: (x,y)→(-y,x) → for A(1,1) → (-1,1), which is not \( A' \)

- 180°: (-1,-1), not \( A' \)

- 270° Clockwise: (x,y)→(-y,x) → (-1,1), not \( A' \)

So the correct answer is 270° Counter-Clockwise? Wait, no, 270° CCW is (y, -x), which for A(1,1) is (1, -1), which matches \( A' \). So yes, 270° CCW is the same as 90° CW. So the rotation is 270° Counter-Clockwise? Wait, no, 270° CCW is rotating 270 degrees counter-clockwise, which is the same as 90 degrees clockwise. So the answer is 270° Counter-Clockwise? Wait, the options:

Option 1: 270° Counter-Clockwise

Option 2: 90° Counter-Clockwise

Option 3: 180°

Option 4: 270° Clockwise

So using the rotation rule, 90° CW is (y, -x), which is 270° CCW. So the correct option is 270° Counter-Clockwise? Wait, no, let's check point B(4,1). 270° CCW: (y, -x) = (1, -4), which is \( B' \) (1, -4). Yes! So B(4,1) → (1, -4) is 270° CCW. So that's correct. So the rotation angle is 270° Counter-Clockwise. Wait, but earlier I thought it was 90° CW, but 270° CCW is the same as 90° CW. So the answer is 270° Counter-Clockwise.

Step2: Confirm with Another Point

Take point D(2,3). 270° CCW rotation: (y, -x) = (3, -2). \( D' \) is at (3, -2), which matches. So this confirms the rotation is 270° Counter-Clockwise.

Answer:

270° Counter-Clockwise