Question

question 10 of 10
what transformations were applied to abcd to obtain abcd?
image of coordinate grid with blue square abcd and red square abcd
a. rotate 90 degrees counterclockwise about the origin, then translate 2 units right.
b. partially visible text

Explanation:

To determine the transformations, we analyze the coordinates (assuming grid details):

Step 1: Identify Rotation

A 90° clockwise rotation about the origin transforms a point \((x, y)\) to \((y, -x)\). For example, if \(A\) is \((0, 5)\) (approx), rotating 90° clockwise gives \((5, 0)\), which aligns with the direction of \(A'\) (negative x - axis, positive y - axis? Wait, no—wait, the red figure is left and down. Wait, let's re - evaluate. A 90° counterclockwise rotation: \((x,y)\to(-y,x)\). Wait, maybe first, let's check the orientation. The original square \(ABCD\) has \(A\) at (0,5), \(B\) at (0,9), \(C\) at (4,9), \(D\) at (4,5) (approx). The red square \(A'B'C'D'\) has \(A'\) at (-8,1), \(B'\) at (-12,1), \(C'\) at (-12,5), \(D'\) at (-8,5) (approx).

Wait, let's do rotation first. Let's take point \(A(0,5)\). A 90° clockwise rotation about the origin: \((x,y)\to(y, - x)\), so \(A(0,5)\to(5,0)\). Then translating left 13 units? No, maybe 90° counterclockwise: \((x,y)\to(-y,x)\), so \(A(0,5)\to(-5,0)\). Then translating left 3 units? Wait, the option B (partially visible) likely is "Rotate 90 degrees clockwise about the origin, then translate 10 units left" (or similar). Wait, the key is:

• Rotation: 90° clockwise (or counterclockwise) changes the orientation. The original square is upright, the red is rotated (sideways to upright? No, the red square has horizontal and vertical sides, same as original, but shifted. Wait, no—wait, the original \(ABCD\) has \(AB\) vertical (from (0,5) to (0,9)), \(AD\) horizontal (from (0,5) to (4,5)). The red \(A'B'C'D'\) has \(A'B'\) horizontal (from (-8,1) to (-12,1)), \(A'D'\) vertical (from (-8,1) to (-8,5)). So the rotation is 90° clockwise (to make \(AB\) horizontal). Then translation: after rotating 90° clockwise, a point like \(A(0,5)\) becomes \((5,0)\), then translating left 13 units? No, the visible option A is "Rotate 90 degrees counterclockwise about the origin, then translate 2 units right"—but that doesn't fit. Wait, the correct transformation (assuming the full option B is "Rotate 90 degrees clockwise about the origin, then translate 10 units left" or similar). But since the problem is about identifying the transformation, let's assume the correct steps:

1. Rotation: 90° clockwise about the origin (changes the orientation of the square so that vertical sides become horizontal).
2. Translation: Then translate left (negative x - direction) by a certain number of units (e.g., 10 units) to align with \(A'B'C'D'\).

(Note: Since the original image’s option B is partially cut, but based on typical transformation problems, the correct transformation is usually a 90° clockwise rotation about the origin followed by a translation left. However, if we assume the visible option A is incorrect, and the intended correct answer involves a 90° clockwise rotation and translation. But since the user’s image shows option A as "Rotate 90 degrees counterclockwise about the origin, then translate 2 units right"—which is incorrect—while the correct transformation is likely "Rotate 90 degrees clockwise about the origin, then translate 10 units left" (or similar, depending on full options). But since the problem is to answer, and assuming the correct option (even with partial text) is the one with 90° clockwise rotation and translation left.

But to match the format, let's proceed.

1. Rotation: The square \(ABCD\) is rotated 90° clockwise about the origin (changing the orientation of its sides).
2. Translation: After rotation, the figure is translated left (in the negative \(x\) - direction) to reach \(A'B'C'D'\).

(Note: Due to partial image, the exact translation units depend on coordinates, but the key transformations are a 90° clockwise rotation about the origin followed by a horizontal translation left. If we assume the correct option is the one with 90° clockwise rotation and translation, e.g., if option B is "Rotate 90 degrees clockwise about the origin, then translate 10 units left", then that’s the answer.

Answer:

(Assuming the correct option is the one with 90° clockwise rotation and translation left; if option B is "Rotate 90 degrees clockwise about the origin, then translate [n] units left", then)
B. Rotate 90 degrees clockwise about the origin, then translate [n] units left (exact units depend on grid, but the rotation type and translation direction match).

(If the full option B is, e.g., "Rotate 90 degrees clockwise about the origin, then translate 10 units left", then the answer is that option.)