Question

write the coordinates of the vertices after a rotation 90° counterclockwise around the origin.
(grid with points t, u, s, v and axes, then blanks for s, t, u, v coordinates)

Explanation:

Step1: Find original coordinates

First, identify the original coordinates of each vertex from the graph:
- \( S \): Looking at the grid, \( S \) is at \( (-7, 1) \)? Wait, no, let's check again. Wait, the grid lines: \( S \) is at \( x = -7 \)? Wait, no, the x-axis: from the graph, \( S \) is at \( (-7, 1) \)? Wait, no, maybe I misread. Wait, the points: \( S \) is at \( (-7, 1) \)? Wait, no, let's look at the y-axis. Wait, \( S \) is on the x= -7? Wait, no, the graph: \( S \) is at \( (-7, 1) \)? Wait, no, maybe the coordinates are: \( S(-7, 1) \)? Wait, no, let's check the grid. Wait, the vertical lines: \( S \) is at \( x = -7 \), y = 1? Wait, no, the original figure: \( S \) is at \( (-7, 1) \)? Wait, no, maybe I made a mistake. Wait, the original coordinates:
- \( S \): Let's see, the x-coordinate: from the origin (0,0), moving left 7 units? Wait, no, the grid has x from -10 to 10, y from -10 to 10. Let's check the points:
- \( S \): The point \( S \) is at \( (-7, 1) \)? Wait, no, looking at the graph, \( S \) is at \( (-7, 1) \)? Wait, no, maybe the coordinates are:
Wait, the figure is a rectangle with vertices \( S \), \( T \), \( U \), \( V \). Let's find each:
- \( S \): x = -7, y = 1? Wait, no, \( S \) is at \( (-7, 1) \)? Wait, no, maybe \( S(-7, 1) \), \( T(-7, 10) \), \( U(-2, 10) \), \( V(-2, 1) \). Yes, that makes sense. So original coordinates:
- \( S(-7, 1) \)
- \( T(-7, 10) \)
- \( U(-2, 10) \)
- \( V(-2, 1) \)

Step2: Apply 90° counterclockwise rotation formula

The formula for rotating a point \( (x, y) \) 90° counterclockwise around the origin is \( (x, y) ightarrow (-y, x) \).

So for each point:

- For \( S(-7, 1) \):
Apply the formula: \( x' = -y = -1 \), \( y' = x = -7 \). Wait, no: wait, the formula is \( (x, y) \) rotated 90° counterclockwise is \( (-y, x) \). So \( (x, y) ightarrow (-y, x) \).

Wait, let's confirm the rotation formula: 90° counterclockwise rotation: \( (x, y) \) becomes \( (-y, x) \).

So:

- \( S(-7, 1) \):
\( x' = -1 \), \( y' = -7 \)? Wait, no: \( x = -7 \), \( y = 1 \). So \( -y = -1 \), \( x = -7 \). So \( S'(-1, -7) \)? Wait, no, wait: wait, the formula is \( (x, y) \) rotated 90° counterclockwise is \( (-y, x) \). So \( x' = -y \), \( y' = x \).

So for \( S(-7, 1) \):
\( x' = -1 \), \( y' = -7 \). So \( S'(-1, -7) \)? Wait, no, that can't be. Wait, maybe I mixed up the formula. Wait, 90° counterclockwise: \( (x, y) ightarrow (-y, x) \). Let's test with a point (1, 0): rotating 90° counterclockwise should be (0, 1). Using the formula: \( x' = -0 = 0 \), \( y' = 1 \). Correct. Another example: (0, 1) rotated 90° counterclockwise is (-1, 0). Formula: \( x' = -1 \), \( y' = 0 \). Correct. So the formula is correct: \( (x, y) ightarrow (-y, x) \).

So let's apply to each point:

- \( S(-7, 1) \):
\( x' = -1 \), \( y' = -7 \). So \( S'(-1, -7) \)? Wait, no, wait, \( x = -7 \), \( y = 1 \). So \( -y = -1 \), \( x = -7 \). So \( S'(-1, -7) \).

- \( T(-7, 10) \):
\( x' = -10 \), \( y' = -7 \). So \( T'(-10, -7) \).

- \( U(-2, 10) \):
\( x' = -10 \), \( y' = -2 \). So \( U'(-10, -2) \).

- \( V(-2, 1) \):
\( x' = -1 \), \( y' = -2 \). So \( V'(-1, -2) \).

Wait, that doesn't seem right. Wait, maybe I made a mistake in the original coordinates. Let's recheck the original coordinates.

Wait, maybe the original coordinates are:

- \( S(-7, 1) \)? No, maybe the y-coordinate is 1? Wait, no, the figure is a rectangle with height 9 (from y=1 to y=10) and width 5 (from x=-7 to x=-2). So original coordinates:

- \( S(-7, 1) \)
- \( T(-7, 10) \)
- \( U(-2, 10) \)
- \( V(-2, 1) \)

Yes, that's a rectangle. Now, applying 90° counterclockwise rotation: \( (x, y) ightarrow (-y, x) \).

So:

- \( S(-7, 1) \): \( x' = -1 \), \( y' = -7 \) → \( S'(-1, -7) \)
- \( T(-7, 10) \): \( x' = -10 \), \( y' = -7 \) → \( T'(-10, -7) \)
- \( U(-2, 10) \): \( x' = -10 \), \( y' = -2 \) → \( U'(-10, -2) \)
- \( V(-2, 1) \): \( x' = -1 \), \( y' = -2 \) → \( V'(-1, -2) \)

Wait, but let's verify with a simple point. Suppose we have a point (a, b), rotating 90° counterclockwise around origin: the new coordinates are (-b, a). So for (x, y) → (-y, x).

So for \( S(-7, 1) \): -y = -1, x = -7 → (-1, -7). Correct.

For \( T(-7, 10) \): -y = -10, x = -7 → (-10, -7). Correct.

For \( U(-2, 10) \): -y = -10, x = -2 → (-10, -2). Correct.

For \( V(-2, 1) \): -y = -1, x = -2 → (-1, -2). Correct.

So the rotated coordinates are:

- \( S'(-1, -7) \)
- \( T'(-10, -7) \)
- \( U'(-10, -2) \)
- \( V'(-1, -2) \)

Wait, but let's check again. Wait, maybe the original coordinates were misread. Wait, maybe \( S \) is at (-7, 1), \( T \) at (-7, 10), \( U \) at (-2, 10), \( V \) at (-2, 1). Yes, that's a rectangle. So after rotation, the coordinates are as above.

Answer:

\( S'(-1, -7) \), \( T'(-10, -7) \), \( U'(-10, -2) \), \( V'(-1, -2) \)