Question

5 what are the coordinates of (-3, 7) after a counterclockwise rotation of 90° about the origin? (25 points) ○ (-3, -7) ○ (7, 3) ○ (3, -7) ○ (-7, -3)

Explanation:

Step1: Recall 90° counterclockwise rotation rule

For a point \((x, y)\), a counterclockwise rotation of \(90^\circ\) about the origin transforms it to \((-y, x)\).

Step2: Apply the rule to \((-3, 7)\)

Here, \(x = -3\) and \(y = 7\). Substituting into the rule, we get \((-y, x)=(-7, -3)\)? Wait, no, wait. Wait, the correct rule for 90° counterclockwise rotation about the origin is \((x,y)\to(-y,x)\)? Wait, no, let's correct. The standard rule: for a point \((x, y)\), rotating 90° counterclockwise about the origin gives \((-y, x)\)? Wait, no, actually, the correct transformation matrix for 90° counterclockwise rotation is \(

$$\begin{pmatrix}0&-1\\1&0\end{pmatrix}$$

\). So applying this to vector \(

$$\begin{pmatrix}x\\y\end{pmatrix}$$

\), we get \(

$$\begin{pmatrix}-y\\x\end{pmatrix}$$

\). Wait, but let's test with a point. Take (1,0), rotating 90° counterclockwise should be (0,1). Using the matrix: \(

$$\begin{pmatrix}0&-1\\1&0\end{pmatrix}$$

$$\begin{pmatrix}1\\0\end{pmatrix}$$

=

$$\begin{pmatrix}0\\1\end{pmatrix}$$

\), correct. Take (0,1), rotating 90° counterclockwise should be (-1,0). Matrix: \(

$$\begin{pmatrix}0&-1\\1&0\end{pmatrix}$$

$$\begin{pmatrix}0\\1\end{pmatrix}$$

=

$$\begin{pmatrix}-1\\0\end{pmatrix}$$

\), correct. So for point \((x,y)\), 90° counterclockwise rotation is \((-y, x)\). Wait, but in our problem, the point is (-3,7). So \(x=-3\), \(y = 7\). Then applying the rule: \((-y, x)=(-7, -3)\)? Wait, no, that's not matching the options. Wait, maybe I mixed up clockwise and counterclockwise. Wait, the rule for 90° clockwise rotation is \((x,y)\to(y, -x)\), and 90° counterclockwise is \((x,y)\to(-y, x)\). Wait, let's check with (-3,7). So \(x=-3\), \(y=7\). Then -y = -7, x = -3? No, that would be (-7, -3), but let's see the options. Wait, maybe I made a mistake. Wait, let's take a point (a,b). Rotating 90° counterclockwise about the origin: the new x-coordinate is -b, new y-coordinate is a. So for (-3,7), new x is -7, new y is -3? But that's option D. Wait, but let's check again. Wait, maybe the rule is different. Wait, another way: when you rotate a point (x,y) 90 degrees counterclockwise around the origin, the coordinates become (-y, x). So (x,y) = (-3,7), so -y = -7, x = -3? Wait, no, x is -3, so the new point is (-7, -3)? But let's check with a simple point. Take (1,2). Rotating 90° counterclockwise: should be (-2,1). Using the rule: -y = -2, x = 1. Correct. So (1,2)→(-2,1). So for (-3,7), -y = -7, x = -3. So (-7, -3). Which is option D. Wait, but let's check the options again. The options are:

A. (-3, -7)

B. (7, 3)

C. (3, -7)

D. (-7, -3)

So according to the rule, the answer should be (-7, -3), which is option D. Wait, but maybe I messed up the rule. Wait, let's think about the quadrants. The original point (-3,7) is in the second quadrant. Rotating 90° counterclockwise would move it to the third quadrant (both x and y negative), which matches (-7, -3). Let's verify with coordinates. The distance from the origin should remain the same. The original distance is \(\sqrt{(-3)^2 + 7^2}=\sqrt{9 + 49}=\sqrt{58}\). The distance of (-7, -3) is \(\sqrt{(-7)^2 + (-3)^2}=\sqrt{49 + 9}=\sqrt{58}\), same distance. So that's correct.

Answer:

D. \((-7, -3)\)