Question

triangle △a′b′c′ is the image of △abc under a rotation about the origin, (0,0).
graph of coordinate plane with triangles abc (blue) and abc (pink)
determine the angle of rotation.
choose 1 answer:

Explanation:

Step1: Identify coordinates of a point and its image

Let's take point \( B \) and its image \( B' \). From the graph, \( B \) has coordinates \( (-3, -1) \)? Wait, no, looking at the grid: \( B \) is at \( (-3, 0) \)? Wait, no, the blue points: \( B \) is at \( (-3, -1) \)? Wait, maybe better to take point \( C \) and \( C' \). Let's check coordinates: \( C \) is at \( (2, 2) \)? Wait, no, the blue \( C \) is at \( (2, 2) \)? Wait, the pink \( C' \) is at \( (2, -1) \)? Wait, maybe I misread. Wait, the origin is (0,0). Let's take point \( B \): blue \( B \) is at \( (-3, -1) \)? No, looking at the x-axis: the blue \( B \) is at \( x=-3 \), y=-1? Wait, maybe better to take a point and its image. Let's take point \( A \): blue \( A \) is at \( (-4, -3) \), pink \( A' \) is at \( (-3, 4) \)? Wait, no, the pink \( A' \) is at \( (-3, 3) \)? Wait, maybe I should look at the rotation direction. Rotation about origin: let's check the angle between vectors \( \overrightarrow{OB} \) and \( \overrightarrow{OB'} \). Let's find coordinates:

Blue \( B \): let's see, the blue triangle: \( B \) is at \( (-3, -1) \)? No, the grid lines: each square is 1 unit. So blue \( B \) is at \( (-3, -1) \)? Wait, no, the blue \( B \) is at \( (-3, 0) \)? Wait, the x-axis: from -7 to 7, y-axis -7 to 7. Let's look at point \( C \): blue \( C \) is at \( (2, 2) \), pink \( C' \) is at \( (2, -1) \)? No, that can't be. Wait, maybe the blue triangle: \( A \) is at \( (-4, -3) \), \( B \) at \( (-3, -1) \), \( C \) at \( (2, 2) \)? No, the pink triangle: \( A' \) at \( (-3, 3) \), \( B' \) at \( (-2, 3) \)? Wait, no, the pink \( A' \) is at \( (-3, 3) \), \( B' \) at \( (-2, 3) \)? Wait, maybe I made a mistake. Let's check the rotation direction. Rotation about origin: if we take a point, say \( B \) (blue) at \( (-3, -1) \), and \( B' \) (pink) at \( (-2, 3) \)? No, maybe better to use the standard rotation angles: 90 degrees, 180 degrees, 270 degrees. Let's check the angle between the vectors from origin to a point and its image.

Wait, let's take point \( A \): blue \( A \) is at \( (-4, -3) \), pink \( A' \) is at \( (-3, 4) \). Let's see the slope of \( OA \): \( \frac{-3}{-4} = \frac{3}{4} \). Slope of \( OA' \): \( \frac{4}{-3} = -\frac{4}{3} \). The product of slopes is \( \frac{3}{4} \times (-\frac{4}{3}) = -1 \), which means the lines are perpendicular (90 degrees). Now, check the direction: from \( OA \) (quadrant III) to \( OA' \) (quadrant II), so it's a 90-degree counterclockwise rotation? Wait, no, quadrant III to quadrant II: if we rotate 90 degrees counterclockwise, a point \( (x, y) \) becomes \( (-y, x) \). Let's test with \( A \): if \( A \) is \( (-4, -3) \), rotating 90 degrees counterclockwise: \( (-(-3), -4) = (3, -4) \)? No, that's not \( A' \). Wait, maybe 90 degrees clockwise: \( (y, -x) \). So \( (-4, -3) \) rotated 90 degrees clockwise: \( (-3, 4) \), which matches \( A' \) (if \( A' \) is at \( (-3, 4) \))? Wait, the pink \( A' \) is at \( (-3, 3) \)? Wait, maybe my coordinate reading is wrong. Let's look again:

Blue triangle: \( A \) is at \( (-4, -3) \) (x=-4, y=-3), \( B \) at \( (-3, -1) \) (x=-3, y=-1), \( C \) at \( (2, 2) \) (x=2, y=2).

Pink triangle: \( A' \) at \( (-3, 3) \) (x=-3, y=3), \( B' \) at \( (-2, 3) \) (x=-2, y=3)? No, the pink \( B' \) is at \( (-2, 3) \)? Wait, the pink \( A' \) is at \( (-3, 3) \), \( B' \) at \( (-2, 3) \), \( C' \) at \( (2, -1) \)? No, this is confusing. Wait, maybe the correct approach is to see the angle between the segments. Let's take point \( B \) (blue) at \( (-3, -1) \) and \( B' \) (pink) at \( (-2, 3) \)? No, maybe the blue \( B \) is at \( (-3, 0) \)? Wait, the x-axis: the blue \( B \) is at \( x=-3 \), y=0? No, the grid has y-axis from -7 to 7. Let's check the origin: the intersection of x and y axes. The blue triangle: \( A \) is at \( (-4, -3) \), \( B \) at \( (-3, -1) \), \( C \) at \( (2, 2) \). The pink triangle: \( A' \) at \( (-3, 3) \), \( B' \) at \( (-2, 3) \), \( C' \) at \( (2, -1) \). Wait, no, maybe the rotation is 90 degrees clockwise. Let's use the rotation formula: 90 degrees clockwise: \( (x, y) ightarrow (y, -x) \). Let's take point \( C \) (blue) at \( (2, 2) \). Rotating 90 degrees clockwise: \( (2, -2) \)? No, that's not \( C' \). Wait, maybe 180 degrees? Rotating 180 degrees: \( (x, y) ightarrow (-x, -y) \). For \( C \) (2,2), 180 degrees rotation would be (-2,-2), which is not \( C' \). Wait, maybe 90 degrees counterclockwise: \( (x, y) ightarrow (-y, x) \). For \( C \) (2,2): (-2, 2), not \( C' \). Wait, maybe I misread the coordinates. Let's look at the graph again:

Blue points: \( A \) is at \( (-4, -3) \) (x=-4, y=-3), \( B \) at \( (-3, -1) \) (x=-3, y=-1), \( C \) at \( (2, 2) \) (x=2, y=2).

Pink points: \( A' \) at \( (-3, 3) \) (x=-3, y=3), \( B' \) at \( (-2, 3) \) (x=-2, y=3), \( C' \) at \( (2, -1) \)? No, that's not matching. Wait, maybe the blue \( C \) is at \( (2, 2) \), pink \( C' \) is at \( (2, -2) \)? No, the pink \( C' \) is at \( (2, -1) \)? Wait, the problem says "rotation about the origin". Let's check the angle between \( \overrightarrow{OC} \) and \( \overrightarrow{OC'} \). If \( C \) is (2,2) and \( C' \) is (2,-2), then the angle is 180 degrees? No, (2,2) to (2,-2) is 180 degrees? Wait, no, (2,2) to (2,-2) is a straight line through the origin, so 180 degrees. But in the graph, the pink \( C' \) is at (2, -1)? Wait, maybe my coordinate reading is wrong. Let's count the grid squares:

For point \( C \) (blue): x=2, y=2 (since from origin, right 2, up 2).

Point \( C' \) (pink): x=2, y=-2? Wait, no, the pink \( C' \) is at (2, -1)? No, the grid lines: each square is 1 unit. So from origin, right 2, down 2: (2, -2). So if \( C \) is (2,2) and \( C' \) is (2,-2), then the rotation is 180 degrees? No, 180 degrees rotation would map (x,y) to (-x,-y). Wait, (2,2) rotated 180 degrees is (-2,-2), not (2,-2). So that's not 180. Wait, maybe 90 degrees clockwise: (2,2) becomes (2,-2)? Wait, no, 90 degrees clockwise rotation formula is (x,y) → (y, -x). So (2,2) → (2, -2). Yes! So (2,2) rotated 90 degrees clockwise is (2, -2). Let's check another point: \( B \) (blue) at (-3, -1). Rotating 90 degrees clockwise: (-1, 3). Wait, is \( B' \) at (-1, 3)? No, the pink \( B' \) is at (-2, 3)? Wait, maybe \( B \) is at (-3, -1), so (x=-3, y=-1). Rotating 90 degrees clockwise: (y, -x) = (-1, 3). So \( B' \) should be at (-1, 3). Let's check the graph: pink \( B' \) is at (-2, 3)? No, maybe my \( B \) coordinate is wrong. Let's take \( A \) (blue) at (-4, -3). Rotating 90 degrees clockwise: (y, -x) = (-3, 4). So \( A' \) should be at (-3, 4). Looking at the graph, pink \( A' \) is at (-3, 3)? No, maybe the graph is slightly different. Wait, the key is that the rotation is 90 degrees clockwise, because the transformation (x,y) → (y, -x) is 90 degrees clockwise. Alternatively, 270 degrees counterclockwise, but 90 degrees clockwise is more likely. Wait, let's check the angle between the vectors. The vector \( \overrightarrow{OA} \) for blue \( A \) is (-4, -3), and \( \overrightarrow{OA'} \) for pink \( A' \) is (-3, 4). The dot product of these vectors is (-4)(-3) + (-3)(4) = 12 - 12 = 0, so they are perpendicular (90 degrees). The cross product (in 2D, the scalar magnitude) is (-4)(4) - (-3)(-3) = -16 - 9 = -25, which is negative, indicating a clockwise rotation (since cross product positive is counterclockwise). So the angle of rotation is 90 degrees clockwise, which is equivalent to 270 degrees counterclockwise, but typically we use the smallest positive angle or the standard direction. Wait, but the options? Wait, the problem says "Choose 1 answer", but the options are not shown. Wait, maybe the options are 90 degrees clockwise, 90 degrees counterclockwise, 180 degrees, 270 degrees. From the dot product being zero (perpendicular) and the cross product negative (clockwise), the angle is 90 degrees clockwise, or 270 degrees counterclockwise. But usually, rotation angle is given as the smallest positive angle, so 90 degrees clockwise, or 270 degrees counterclockwise. But in the graph, looking at the positions, the blue triangle is in quadrants III and I, and the pink triangle is in quadrants II and IV. So rotating 90 degrees clockwise would move from III to II, etc. So the angle of rotation is 90 degrees clockwise, or 270 degrees counterclockwise. But likely, the answer is 90 degrees clockwise, or 90 degrees (assuming clockwise is the direction). Wait, maybe the options are 90 degrees, 180 degrees, 270 degrees. So the correct angle is 90 degrees clockwise, which is 90 degrees.

Step2: Confirm with rotation formula

For a point \( (x, y) \), rotating 90 degrees clockwise about the origin gives \( (y, -x) \). Let's test with point \( A \): if \( A \) is \( (-4, -3) \), then \( A' \) should be \( (-3, 4) \). Looking at the graph, pink \( A' \) is at \( (-3, 3) \)? Wait, maybe my coordinate reading is off. Alternatively, maybe the rotation is 90 degrees counterclockwise, which is \( (-y, x) \). For \( A(-4, -3) \), that would be \( (3, -4) \), which doesn't match. Wait, maybe the correct coordinates are:

Blue \( A \): (-4, -3), pink \( A' \): (-3, 4) – yes, that's (y, -x) where x=-4, y=-3: y=-3, -x=4, so (-3, 4). Yes, that matches. So the rotation is 90 degrees clockwise, so the angle of rotation is 90 degrees.

Answer:

90 degrees (clockwise) or 270 degrees (counterclockwise), but typically 90 degrees is the answer here.