the figure below has a point marked with a large dot. first, rotate the figure 90° clockwise about the…

Explanation:

Step1: Find original coordinates

Locate the marked point on the grid. From the x - axis (horizontal) and y - axis (vertical), the original point is at (x=-2), (y = 4), so original coordinates are ((-2,4)).

Step2: Apply 90° clockwise rotation rule

The rule for rotating a point ((x,y)) 90° clockwise about the origin is ((x,y)\to(y, - x)). Substitute (x=-2) and (y = 4) into the rule: new (x = 4), new (y=-(-2)=2)? Wait, no, correction: the correct rule for 90° clockwise rotation about the origin is ((x,y)\to(y,-x)). So for ((-2,4)), (x=-2), (y = 4), so the new point is ((4,2))? Wait, let's re - check the grid. Wait, maybe I misread the original coordinates. Let's look at the grid again. The x - axis: moving from left to right, the origin is at (0,0). The marked point: let's count the units. From the origin (0,0), moving left 2 units on x (so x=-2) and up 4 units on y (so y = 4). Now, 90° clockwise rotation: the formula is ((x,y)) becomes ((y, - x)). So (x=-2), (y = 4), so (y = 4), (-x=2). So the rotated point is ((4,2))? Wait, maybe I made a mistake. Wait, another way: 90° clockwise rotation is equivalent to 270° counter - clockwise. The standard rotation matrices: for 90° clockwise, the transformation matrix is (\begin{pmatrix}0&1\-1&0\end{pmatrix}). So for a vector (\begin{pmatrix}x\y\end{pmatrix}), the new vector is (\begin{pmatrix}0&1\-1&0\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}y\-x\end{pmatrix}). So for ((-2,4)), the new point is (\begin{pmatrix}4\2\end{pmatrix}), so ((4,2)). Wait, but let's check the grid again. Wait, maybe the original x - coordinate was misread. Wait, looking at the grid, the point is at x=-2 (since from 0, moving left 2) and y = 4 (moving up 4). So original is (-2,4). After 90° clockwise rotation, using ((x,y)\to(y,-x)), we get (4,2). Wait, but maybe I messed up the sign. Wait, no: if (x=-2), then (-x = 2), so (y, - x) is (4,2).

Wait, maybe the original coordinates were ( - 2,4). Let's confirm the grid. The x - axis: the numbers on the x - axis, from left: - 8, - 7, - 6, - 5, - 4, - 3, - 2, - 1, 0, 1, 2,... The y - axis: from bottom: - 8, - 7,...0,1,2,3,4,5,6,... The marked point is at x=-2 (since it's 2 units left of origin) and y = 4 (4 units up). So original is (-2,4). Rotated 90° clockwise: (4,2). Wait, but let's check with another approach. Let's take a point (0,1), rotate 90° clockwise, it should go to (1,0), which fits (y,-x) where x = 0, y = 1: (1,0). Another example: (1,0) rotated 90° clockwise: (0, - 1), which is (y,-x) with x = 1, y = 0: (0, - 1). So the rule holds. So for (-2,4), rotated 90° clockwise: (4,2).

Answer:

Point in original figure: ((-2,4)) Point in final figure: ((4,2))