Question

question 5
10 pts
g.gsr.3.2 (mc)
if parallelogram abcd were rotated 270° counter - clockwise around the origin, where would c’ be located?
image of a coordinate grid with parallelogram abcd
options: (-2, -4), (4, 2), (4, -2), (-2, 4)

Explanation:

Step1: Find Coordinates of C

From the graph, point C is at \((-4, -2)\).

Step2: Apply 270° CCW Rotation Rule

The rule for rotating a point \((x, y)\) 270° counter - clockwise about the origin is \((x, y)\to(y, -x)\).
For point C \((-4, -2)\), substitute \(x = - 4\) and \(y=-2\) into the rule.
We get \(y=-2\) and \(-x = -(-4)=4\). So the coordinates of \(C'\) are \((-2,4)\)? Wait, no, wait. Wait, the rotation rule for 270° counter - clockwise is \((x,y)\to(y, - x)\). Wait, let's re - check. The standard rotation rules:

• 90° counter - clockwise: \((x,y)\to(-y,x)\)
• 180° counter - clockwise: \((x,y)\to(-x,-y)\)
• 270° counter - clockwise: \((x,y)\to(y, - x)\)

Wait, point C is \((-4, - 2)\). So \(x=-4\), \(y = - 2\). Then applying 270° counter - clockwise rotation: \((y,-x)=(-2, -(-4))=(-2,4)\)? Wait, no, that seems wrong. Wait, maybe I mixed up the rule. Wait, another way: rotating 270° counter - clockwise is the same as rotating 90° clockwise. The rule for 90° clockwise is \((x,y)\to(y, - x)\). Wait, let's take a point. Let's take a simple point, say (1,0). Rotating 270° counter - clockwise around the origin: (1,0) rotated 270° counter - clockwise. 90°: (- 0,1)=(0,1); 180°: (-1,0); 270°: (0, - 1). Using the rule \((x,y)\to(y, - x)\): (0, - 1), which is correct. Another point: (0,1). Rotating 270° counter - clockwise: (1,0). Using the rule \((x,y)\to(y, - x)\): (1, - 0)=(1,0), correct. So the rule is correct.

Wait, point C is at (-4, - 2). So \(x=-4\), \(y = - 2\). Then \(y=-2\), \(-x = 4\). So the new point is \((-2,4)\)? Wait, no, wait \((y, - x)\) where \(x=-4\), \(y=-2\) is \((-2,4)\). Let's check the options. One of the options is (-2,4). Wait, but let's re - check the coordinates of C. From the graph, looking at the grid, point C is at x=-4, y=-2. So yes, (-4,-2). Then applying 270° counter - clockwise rotation: (y, - x)=(-2,4). So the coordinates of C' are (-2,4).

Answer:

\(\boldsymbol{(-2, 4)}\) (corresponding to the option D. (-2, 4) if we assume the last option is D. (-2,4))