Question

find the image of (9, -1) under the transformation of ( r_{90} circ t_{langle 2, 3
angle} ).

Explanation:

Step1: Identify the transformations

Assume \( T_{\langle a,b
angle} \) is a translation and \( R_{90^\circ} \) is a 90° rotation (counterclockwise, usually). But the translation vector \( \langle a,b
angle \) is not clear from the image. Wait, maybe the translation is \( T_{\langle 2,3
angle} \) (assuming a common typo or missing part). Let's first handle translation then rotation.

Step 1: Translation \( T_{\langle h,k

angle} \)
If translation is \( T_{\langle 2,3
angle} \) (assuming \( T_{\langle 2,3
angle} \) meaning add (2,3) to the point), then for point (9, -1):
New point after translation: \( (9 + 2, -1 + 3) = (11, 2) \)

Step 2: Rotation \( R_{90^\circ} \) (counterclockwise)

The formula for 90° counterclockwise rotation about the origin is \( (x,y) \to (-y, x) \)
Applying to (11, 2): \( (-2, 11) \)

Wait, maybe the translation is different. If the translation is \( T_{\langle -2, -3
angle} \) or the rotation is clockwise. Wait, the problem says \( R_{90} \circ T \), so we first do translation \( T \), then rotation \( R_{90} \).

But since the image is unclear, maybe the translation is \( T_{\langle 2,3
angle} \) (common in such problems). Let's recheck.

Alternative: Maybe the translation is \( T_{\langle -2, -3
angle} \), but no, let's assume the translation vector is \( \langle 2,3
angle \) (adding 2 to x, 3 to y).

Wait, maybe the original problem has translation \( T_{\langle 2,3
angle} \) (move 2 right, 3 up). So:

1. Translate (9, -1) by \( \langle 2,3

angle \): \( (9 + 2, -1 + 3) = (11, 2) \)

1. Rotate 90° counterclockwise: \( (x,y) \to (-y, x) \), so (11, 2) becomes (-2, 11)

Or if rotation is clockwise, formula is \( (x,y) \to (y, -x) \), then (11,2) becomes (2, -11)

But without clear translation vector, it's hard. Wait, maybe the translation is \( T_{\langle -2, -3
angle} \). Let's try:

1. Translate (9, -1) by \( \langle -2, -3

angle \): \( (9 - 2, -1 - 3) = (7, -4) \)

1. Rotate 90° counterclockwise: (-(-4), 7) = (4, 7) (wait, no: 90° counterclockwise is (-y, x), so (7, -4) becomes (4, 7)? Wait, no: (x,y)=(7,-4), so -y = 4, x=7, so (4,7).

But this is guesswork. Alternatively, maybe the translation is \( T_{\langle 0,0
angle} \) (no translation), but that's unlikely.

Wait, the problem's image is partially visible. Let's assume the translation is \( T_{\langle 2,3
angle} \) (as a common problem). Then:

After translation: (9+2, -1+3)=(11,2)

After 90° counterclockwise rotation: (-2,11)

So the image of (9, -1) under \( R_{90} \circ T \) (T is translation by (2,3)) is (-2, 11)

Answer:

\boxed{(-2, 11)} (assuming translation \( T_{\langle 2,3
angle} \) and 90° counterclockwise rotation)