Question

11 triangle abc was rotated about the origin to create triangle abc. which of the following describe this transformation? move the checkmarks to select three answers. a 90° counterclockwise rotation a 180° clockwise rotation a 270° clockwise rotation (x, y) → (-x, -y) (x, y) → (-y, x) (x, y) → (y, -x)

Explanation:

Step1: Identify coordinates of original and rotated points

First, find coordinates of \( A, B, C \) and \( A', B', C' \). From the graph:
- \( A(-4, 3) \), \( B(-3, 4) \), \( C(-1, 2) \)
- \( A'(-3, -5) \)? Wait, no, looking again: Wait, \( A' \) is at \( (-3, -5) \)? Wait, no, let's re - check. Wait, the rotated triangle \( A'B'C' \): \( A' \) seems to be \( (-3, -5) \)? Wait, no, maybe I misread. Wait, actually, let's take a point, say \( A(-4, 3) \), \( A' \) should be such that after rotation. Wait, let's check the transformation rules.

Step2: Analyze rotation rules

- **180° rotation rule**: For a 180° clockwise (or counter - clockwise, since 180° rotation is same in both directions) rotation about the origin, the rule is \( (x,y)\to(-x,-y) \). Let's test with point \( A(-4,3) \): \( -x = 4 \), \( -y=-3 \)? No, that's not matching. Wait, maybe I made a mistake in coordinates. Wait, looking at the graph again: \( A \) is at \( (-4, 3) \), \( B \) at \( (-3, 4) \), \( C \) at \( (-1, 2) \). \( A' \) is at \( (-3, -5) \)? No, wait, the lower triangle: \( A' \) is at \( (-3, -5) \)? Wait, no, maybe the coordinates are: \( A(-4,3) \), \( A'(-3, -5) \)? No, that can't be. Wait, maybe I should check the 270° clockwise rotation rule. The rule for 270° clockwise rotation (or 90° counter - clockwise) is \( (x,y)\to(-y,x) \)? No, wait, the rule for 90° counter - clockwise is \( (x,y)\to(-y,x) \), 90° clockwise is \( (x,y)\to(y, - x) \), 180° is \( (x,y)\to(-x,-y) \), 270° clockwise is \( (x,y)\to(-y,x) \)? Wait, no, let's recall:

- 90° counter - clockwise: \( (x,y)\to(-y,x) \)
- 90° clockwise: \( (x,y)\to(y, - x) \)
- 180°: \( (x,y)\to(-x,-y) \)
- 270° clockwise (which is equivalent to 90° counter - clockwise): \( (x,y)\to(-y,x) \)
- 270° counter - clockwise (equivalent to 90° clockwise): \( (x,y)\to(y, - x) \)

Wait, let's take point \( A(-4,3) \). Let's apply 270° clockwise rotation (rule \( (x,y)\to(-y,x) \)): \( -y=-3 \), \( x = - 4 \)? No, that's \( (-3,-4) \), not matching. Wait, maybe the correct rotation is 180°? Wait, no, let's check point \( B(-3,4) \). If we apply 180° rotation: \( (-x,-y)=(3, - 4) \), but \( B' \) is at \( (-4, - 3) \)? Wait, I think I misread the coordinates. Let's re - extract coordinates:

Looking at the grid:

- Original triangle \( ABC \):
- \( A \): x = - 4, y = 3 (so \( A(-4,3) \))
- \( B \): x = - 3, y = 4 (so \( B(-3,4) \))
- \( C \): x = - 1, y = 2 (so \( C(-1,2) \))

- Rotated triangle \( A'B'C' \):
- \( A' \): x = - 3, y = - 5? No, wait, the lower triangle: \( A' \) is at ( - 3, - 5)? No, maybe the y - axis is flipped? Wait, no, the grid has positive y up, negative y down. Wait, \( A' \) is at ( - 3, - 5)? No, that doesn't make sense. Wait, maybe the correct coordinates for \( A' \) are (3, - 3)? No, I think I made a mistake. Let's use the transformation rule \( (x,y)\to(y, - x) \) (270° counter - clockwise or 90° clockwise). For \( A(-4,3) \): \( y = 3 \), \( -x = 4 \), so \( (3,4) \)? No. Wait, the correct approach is:

Let's check the three correct options. The three correct options should be:

1. A 270° clockwise rotation (equivalent to 90° counter - clockwise)
2. A 90° counter - clockwise rotation (same as 270° clockwise)
3. The transformation \( (x,y)\to(-y,x) \) (rule for 90° counter - clockwise/270° clockwise)

Wait, no, let's check the 180° rotation: \( (x,y)\to(-x,-y) \). For \( A(-4,3) \), \( -x = 4 \), \( -y=-3 \), so \( (4,-3) \), not matching. Wait, maybe the coordinates are:

Wait, \( A(-4,3) \), \( A'(-3, - 5) \) is wrong. Let's look at the graph again. The original triangle is in the second quadrant (x negative, y positive), the rotated triangle is in the third quadrant? No, the rotated triangle \( A'B'C' \): \( A' \) is at ( - 3, - 5)? No, the y - coordinate of \( A' \) is - 5? Wait, the grid lines: each square is 1 unit. So \( A \) is at ( - 4, 3) (4 units left of origin, 3 up), \( B \) at ( - 3, 4) (3 left, 4 up), \( C \) at ( - 1, 2) (1 left, 2 up). The rotated triangle: \( A' \) is at ( - 3, - 5)? No, that's too far down. Wait, maybe \( A' \) is at (3, - 3)? No, I think I messed up. Let's use the transformation rules.

The three correct options are:

- A 90° counterclockwise rotation (rule \( (x,y)\to(-y,x) \))
- A 270° clockwise rotation (same as 90° counterclockwise, rule \( (x,y)\to(-y,x) \))
- The transformation \( (x,y)\to(-y,x) \)

Wait, no, let's check with a point. Take \( C(-1,2) \). After 90° counterclockwise rotation, \( (x,y)\to(-y,x) \), so \( (-2,-1) \). Looking at \( C' \), if \( C' \) is at ( - 2, - 1), that matches. Let's check \( B(-3,4) \): \( (-4,-3) \), and \( B' \) should be at ( - 4, - 3), which matches the graph. \( A(-4,3) \): \( (-3,-4) \), and \( A' \) is at ( - 3, - 4)? Wait, maybe I misread the coordinates. So:

- 90° counterclockwise rotation: uses rule \( (x,y)\to(-y,x) \)
- 270° clockwise rotation: is equivalent to 90° counterclockwise rotation (since 270° clockwise = 360° - 90° clockwise = 90° counterclockwise)
- The transformation \( (x,y)\to(-y,x) \)

Wait, but also, the 180° rotation is \( (x,y)\to(-x,-y) \), which doesn't match. Wait, maybe the correct three options are:

1. A 90° counterclockwise rotation
2. A 270° clockwise rotation
3. \( (x,y)\to(-y,x) \)

Wait, let's confirm the rotation rules:

- 90° counterclockwise: \( (x,y)\to(-y,x) \)
- 90° clockwise: \( (x,y)\to(y, - x) \)
- 180°: \( (x,y)\to(-x,-y) \)
- 270° clockwise: \( (x,y)\to(-y,x) \) (same as 90° counterclockwise)
- 270° counterclockwise: \( (x,y)\to(y, - x) \) (same as 90° clockwise)

So, for the given triangle, when we apply \( (x,y)\to(-y,x) \), let's check \( A(-4,3) \): \( -y=-3 \), \( x = - 4 \)? No, wait, \( (x,y)=(-4,3) \), so \( -y=-3 \), \( x=-4 \)? No, that's \( (-3,-4) \)? Wait, \( x=-4 \), \( y = 3 \), so \( -y=-3 \), \( x=-4 \)? No, the rule is \( (x,y)\to(-y,x) \), so \( x=-4 \), \( y = 3 \), so \( -y=-3 \), \( x=-4 \)? No, that's \( (-3,-4) \). Let's check the coordinates of \( A' \) in the graph. If \( A' \) is at \( (-3,-4) \), then it matches. \( B(-3,4) \): \( -y=-4 \), \( x=-3 \), so \( (-4,-3) \), which matches \( B' \). \( C(-1,2) \): \( -y=-2 \), \( x=-1 \), so \( (-2,-1) \), which matches \( C' \).

Now, let's check the options:

- "A 90° counterclockwise rotation": correct, since 90° counterclockwise uses \( (x,y)\to(-y,x) \)
- "A 180° clockwise rotation": incorrect, since 180° uses \( (x,y)\to(-x,-y) \), which would give \( A(4,-3) \), not matching
- "A 270° clockwise rotation": correct, since 270° clockwise is equivalent to 90° counterclockwise (360 - 270 = 90 counterclockwise? Wait, no: 270° clockwise rotation: the formula is \( (x,y)\to(-y,x) \), same as 90° counterclockwise. Because rotating 270° clockwise is the same as rotating 90° counterclockwise (since 270 = 360 - 90)
- " \( (x,y)\to(-x,-y) \)": incorrect, as shown
- " \( (x,y)\to(-y,x) \)": correct, this is the rule for 90° counterclockwise/270° clockwise rotation
- " \( (x,y)\to(y, - x) \)": incorrect, this is the rule for 90° clockwise/270° counterclockwise rotation

Answer:

A. A 90° counterclockwise rotation
C. A 270° clockwise rotation
E. \( (x, y) \to (-y, x) \)