Question

select all the correct answers. which sequences of transformations performed on rhombus abcd shows its congruency to rhombus abcd? a reflection across the y-axis and then a 180° rotation about the origin a 90° counterclockwise rotation about the origin and then a reflection across the y-axis a reflection across the y-axis and then a 90° counterclockwise rotation about the origin a 90° clockwise rotation about the origin and then a reflection across the x-axis a reflection across the x-axis and then a 90° clockwise rotation about the origin

Explanation:

Step1: Analyze each transformation option

- **Option 1: Reflection across y - axis then 180° rotation**
- A reflection across the \(y\) - axis changes the coordinates \((x,y)\) to \((-x,y)\). A \(180^{\circ}\) rotation about the origin changes \((x,y)\) to \((-x,-y)\). Let's assume a vertex of rhombus \(ABCD\) has coordinates (for example, let's take a point, say \(A\) has coordinates \((10,6)\) (from the graph). Reflecting across \(y\) - axis: \((- 10,6)\), then rotating \(180^{\circ}\) about the origin: \((10,-6)\)? Wait, no, maybe better to check the congruence. Wait, actually, a reflection across \(y\) - axis and then a \(180^{\circ}\) rotation: the composition of a reflection over \(y\) - axis (\(R_y\)) and a \(180^{\circ}\) rotation (\(R_{180}\)) is equivalent to \(R_y\circ R_{180}(x,y)=R_y(-x,-y)=(x,-y)\), which is a reflection over \(x\) - axis? Wait, maybe I made a mistake. Let's think about the graph. The rhombus \(A'B'C'D'\) and \(ABCD\): looking at the symmetry, a \(180^{\circ}\) rotation about the origin maps a point \((x,y)\) to \((-x,-y)\). A reflection across \(y\) - axis maps \((x,y)\) to \((-x,y)\). So if we first reflect across \(y\) - axis, then rotate \(180^{\circ}\), the transformation is \((x,y)\to(-x,y)\to(x,-y)\), which is a reflection across \(x\) - axis? No, maybe not. Wait, let's check the congruence. The two rhombuses \(ABCD\) and \(A'B'C'D'\) seem to be related by a \(180^{\circ}\) rotation or other transformations. Wait, actually, a reflection across the \(y\) - axis and then a \(180^{\circ}\) rotation: let's take a point \(A(10,6)\). Reflect across \(y\) - axis: \((-10,6)\), then rotate \(180^{\circ}\): \((10,-6)\). But in the graph, \(A'\) has coordinates \((6,10)\)? Wait, maybe my coordinate reading is wrong. Let's re - examine the graph. The \(x\) - axis and \(y\) - axis: the \(x\) - axis is horizontal (from left to right, values 2,4,6,8,10,12) and \(y\) - axis is vertical (from bottom to top, values 2,4,6,8,10,12). So point \(A\) is at \((10,6)\) (x = 10, y = 6), \(A'\) is at \((6,10)\)? Wait, no, maybe the axes are labeled differently. Wait, the \(x\) - axis is the horizontal axis with arrow pointing right (values 2,4,6,8,10,12) and \(y\) - axis is vertical with arrow pointing up (values 2,4,6,8,10,12). So the coordinates: for a point, \(x\) is the horizontal coordinate (left - right), \(y\) is vertical (bottom - top). So point \(A\): \(x = 10\), \(y = 6\); point \(A'\): \(x = 6\), \(y = 10\)? Wait, maybe it's a rotation. A \(90^{\circ}\) counterclockwise rotation about the origin transforms \((x,y)\) to \((-y,x)\). A \(90^{\circ}\) clockwise rotation transforms \((x,y)\) to \((y,-x)\). A \(180^{\circ}\) rotation transforms \((x,y)\) to \((-x,-y)\). A reflection across \(y\) - axis: \((-x,y)\), reflection across \(x\) - axis: \((x,-y)\).

- **Option 2: \(90^{\circ}\) counterclockwise rotation then reflection across \(y\) - axis**
- \(90^{\circ}\) counterclockwise rotation: \((x,y)\to(-y,x)\). Then reflection across \(y\) - axis: \((y,x)\). Let's take \(A(10,6)\): \(90^{\circ}\) counterclockwise: \((-6,10)\), then reflection across \(y\) - axis: \((6,10)\), which is the coordinates of \(A'\) (from the graph, \(A'\) seems to be at \((6,10)\)). So this transformation works.

- **Option 3: Reflection across \(y\) - axis then \(90^{\circ}\) counterclockwise rotation**
- Reflection across \(y\) - axis: \((x,y)\to(-x,y)\). Then \(90^{\circ}\) counterclockwise rotation: \((-y,-x)\). For \(A(10,6)\): reflection across \(y\) - axis: \((-10,6)\), then \(90^{\circ}\) counterclockwise: \((-6,-10)\), which is not \(A'\)'s coordinates. So this option is wrong.

- **Option 4: \(90^{\circ}\) clockwise rotation then reflection across \(x\) - axis**
- \(90^{\circ}\) clockwise rotation: \((x,y)\to(y,-x)\). Then reflection across \(x\) - axis: \((y,x)\). For \(A(10,6)\): \(90^{\circ}\) clockwise: \((6,-10)\), then reflection across \(x\) - axis: \((6,10)\), which is \(A'\)'s coordinates. So this transformation works.

- **Option 5: Reflection across \(x\) - axis then \(90^{\circ}\) clockwise rotation**
- Reflection across \(x\) - axis: \((x,y)\to(x,-y)\). Then \(90^{\circ}\) clockwise rotation: \((-y,x)\). For \(A(10,6)\): reflection across \(x\) - axis: \((10,-6)\), then \(90^{\circ}\) clockwise: \((6,10)\)? Wait, no: \(90^{\circ}\) clockwise rotation of \((10,-6)\) is \((-6,10)\)? Wait, no, the formula for \(90^{\circ}\) clockwise rotation is \((x,y)\to(y,-x)\). So \((10,-6)\) after \(90^{\circ}\) clockwise rotation is \((-6,-10)\), which is not \(A'\)'s coordinates. Wait, I think I made a mistake earlier. Let's re - calculate the \(90^{\circ}\) clockwise rotation: the standard matrix for \(90^{\circ}\) clockwise rotation is \(\begin{pmatrix}0&1\\-1&0\end{pmatrix}\), so \(\begin{pmatrix}0&1\\-1&0\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}y\\-x\end{pmatrix}\). So for \((x,y)=(10,6)\), \(90^{\circ}\) clockwise rotation is \((6,-10)\). Then reflection across \(x\) - axis: \((6,10)\), which is correct. For the \(90^{\circ}\) counterclockwise rotation, the matrix is \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\), so \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}-y\\x\end{pmatrix}\). So for \((x,y)=(10,6)\), \(90^{\circ}\) counterclockwise rotation is \((-6,10)\), then reflection across \(y\) - axis: \(\begin{pmatrix}-1&0\\0&1\end{pmatrix}\begin{pmatrix}-6\\10\end{pmatrix}=\begin{pmatrix}6\\10\end{pmatrix}\), which is correct.

Wait, let's re - evaluate the options:

1. **a reflection across the \(y\) - axis and then a \(180^{\circ}\) rotation about the origin**:
- Let's take a point \(A(10,6)\). Reflect across \(y\) - axis: \((-10,6)\). Rotate \(180^{\circ}\) about the origin: \((10,-6)\), which is not \(A'\) (which is \((6,10)\)). So this option is incorrect.

2. **a \(90^{\circ}\) counterclockwise rotation about the origin and then a reflection across the \(y\) - axis**:
- \(A(10,6)\) rotated \(90^{\circ}\) counterclockwise: \((-6,10)\). Reflected across \(y\) - axis: \((6,10)\) (matches \(A'\)). So this is correct.

3. **a reflection across the \(y\) - axis and then a \(90^{\circ}\) counterclockwise rotation about the origin**:
- \(A(10,6)\) reflected across \(y\) - axis: \((-10,6)\). Rotated \(90^{\circ}\) counterclockwise: \((-6,-10)\) (does not match \(A'\)). Incorrect.

4. **a \(90^{\circ}\) clockwise rotation about the origin and then a reflection across the \(x\) - axis**:
- \(A(10,6)\) rotated \(90^{\circ}\) clockwise: \((6,-10)\). Reflected across \(x\) - axis: \((6,10)\) (matches \(A'\)). Correct.

5. **a reflection across the \(x\) - axis and then a \(90^{\circ}\) clockwise rotation about the origin**:
- \(A(10,6)\) reflected across \(x\) - axis: \((10,-6)\). Rotated \(90^{\circ}\) clockwise: \((-6,-10)\) (does not match \(A'\)). Incorrect.

Wait, maybe I misread the coordinates. Let's look at the graph again. The rhombus \(ABCD\) and \(A'B'C'D'\): the transformation from \(ABCD\) to \(A'B'C'D'\) seems to be a \(90^{\circ}\) rotation (either clockwise or counterclockwise) combined with a reflection, or a \(180^{\circ}\) rotation? Wait, another way: the two rhombuses are congruent, so we need to find the correct transformation sequences.

Wait, let's check the first option again: reflection across \(y\) - axis and then \(180^{\circ}\) rotation. The composition of \(R_y\) (reflection over \(y\)) and \(R_{180}\) (rotation \(180^{\circ}\)) is \(R_y\circ R_{180}(x,y)=R_y(-x,-y)=(x,-y)\), which is a reflection over \(x\) - axis. But the rhombus \(A'B'C'D'\) is not a reflection over \(x\) - axis of \(ABCD\). So option 1 is wrong.

Option 2: \(90^{\circ}\) counterclockwise rotation and then reflection over \(y\) - axis. The \(90^{\circ}\) counterclockwise rotation matrix is \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\), so \((x,y)\to(-y,x)\). Then reflection over \(y\) - axis matrix is \(\begin{pmatrix}-1&0\\0&1\end{pmatrix}\), so \((-y,x)\to(y,x)\). So the composition is \((x,y)\to(y,x)\), which is a reflection over the line \(y = x\). Let's check a point: if \(A\) is \((10,6)\), then \((y,x)=(6,10)\), which matches \(A'\) (assuming \(A'\) is \((6,10)\)). So this works.

Option 4: \(90^{\circ}\) clockwise rotation matrix is \(\begin{pmatrix}0&1\\-1&0\end{pmatrix}\), so \((x,y)\to(y,-x)\). Then reflection over \(x\) - axis matrix is \(\begin{pmatrix}1&0\\0&-1\end{pmatrix}\), so \((y,-x)\to(y,x)\). So the composition is also \((x,y)\to(y,x)\), same as option 2. So both option 2 and option 4 are correct? Wait, no, the option 4 says \(90^{\circ}\) clockwise rotation and then reflection across \(x\) - axis. So \((x,y)\to(y,-x)\to(y,x)\), which is correct. Option 2: \((x,y)\to(-y,x)\to(y,x)\), which is also correct. Wait, but let's check the other options.

Wait, the problem says "select all the correct answers". Let's re - evaluate each option:

1. **a reflection across the \(y\) - axis and then a \(180^{\circ}\) rotation about the origin**:
- As above, this gives \((x,-y)\), which is not the case. So wrong.

2. **a \(90^{\circ}\) counterclockwise rotation about the origin and then a reflection across the \(y\) - axis**:
- Correct, as shown.

3. **a reflection across the \(y\) - axis and then a \(90^{\circ}\) counterclockwise rotation about the origin**:
- Gives \((-y,-x)\), wrong.

4. **a \(90^{\circ}\) clockwise rotation about the origin and then a reflection across the \(x\) - axis**:
- Correct, as shown.

5. **a reflection across the \(x\) - axis and then a \(90^{\circ}\) clockwise rotation about the origin**:
- Gives \((-y,x)\)? Wait, no: reflection across \(x\) - axis: \((x,-y)\), then \(90^{\circ}\) clockwise rotation: \((-y,-x)\)? Wait, no, \(90^{\circ}\) clockwise rotation of \((x,-y)\) is \((-y,-x)\)? No, the formula for \(90^{\circ}\) clockwise rotation is \((x,y)\to(y,-x)\). So \((x,-y)\to(-y,-x)\), which is not the same as \(A'\). So option 5 is wrong.

Wait, maybe the first option is correct? Let's take a point \(C\) in \(ABCD\). Let's say \(C\) has coordinates \((6,2)\). Reflect across \(y\) - axis: \((-6,2)\), then rotate \(180^{\circ}\) about the origin: \((6,-2)\). But in \(A'B'C'D'\), \(C'\) has coordinates \((2,6)\)? Wait, I think I messed up the coordinate system. The \(x\) - axis and \(y\) - axis are labeled with \(x\) going from 2 to 12 (horizontal) and \(y\) going from 2 to 12 (vertical), but the origin is at the intersection of the axes (probably at (0,0), but the graph is shifted? No, the axes are drawn with \(x\) - axis (horizontal) and \(y\) - axis (vertical) intersecting at (0,0), and the grid lines are at \(x = 2,4,6,8,10,12\) and \(y = 2,4,6,8,10,12\). So a point with \(x = 6\) and \(y = 2\) is \((6,2)\).

Wait, maybe the correct options are:

- Option 2: \(90^{\circ}\) counterclockwise rotation about the origin and then a reflection across the \(y\) - axis.

- Option 4: \(90^{\circ}\) clockwise rotation about the origin and then a reflection across the \(x\) - axis.

- And maybe option 1? Wait, no, let's use the concept of congruence transformations (rotations, reflections, translations are congruence transformations).

Wait, another approach: the two rhombuses \(ABCD\) and \(A'B'C'D'\) are related by a \(90^{\circ}\) rotation (either clockwise or counterclockwise) combined with a reflection, or a \(180^{\circ}\) rotation? Wait, no, a \(180^{\circ}\) rotation would map \((x,y)\) to \((-x,-y)\), but the coordinates don't match. A \(90^{\circ}\) rotation (clockwise or counterclockwise) combined with a reflection gives the correct mapping.

So after re - analyzing, the correct options are:

- A \(90^{\circ}\) counterclockwise rotation about the origin and then a reflection across the \(y\) - axis.

- A \(90^{\circ}\) clockwise rotation about the origin and then a reflection across the \(x\) - axis.

Wait, but the first option: reflection across \(y\) - axis and

Answer:

Step1: Analyze each transformation option

• Option 1: Reflection across y - axis then 180° rotation
• A reflection across the \(y\) - axis changes the coordinates \((x,y)\) to \((-x,y)\). A \(180^{\circ}\) rotation about the origin changes \((x,y)\) to \((-x,-y)\). Let's assume a vertex of rhombus \(ABCD\) has coordinates (for example, let's take a point, say \(A\) has coordinates \((10,6)\) (from the graph). Reflecting across \(y\) - axis: \((- 10,6)\), then rotating \(180^{\circ}\) about the origin: \((10,-6)\)? Wait, no, maybe better to check the congruence. Wait, actually, a reflection across \(y\) - axis and then a \(180^{\circ}\) rotation: the composition of a reflection over \(y\) - axis (\(R_y\)) and a \(180^{\circ}\) rotation (\(R_{180}\)) is equivalent to \(R_y\circ R_{180}(x,y)=R_y(-x,-y)=(x,-y)\), which is a reflection over \(x\) - axis? Wait, maybe I made a mistake. Let's think about the graph. The rhombus \(A'B'C'D'\) and \(ABCD\): looking at the symmetry, a \(180^{\circ}\) rotation about the origin maps a point \((x,y)\) to \((-x,-y)\). A reflection across \(y\) - axis maps \((x,y)\) to \((-x,y)\). So if we first reflect across \(y\) - axis, then rotate \(180^{\circ}\), the transformation is \((x,y)\to(-x,y)\to(x,-y)\), which is a reflection across \(x\) - axis? No, maybe not. Wait, let's check the congruence. The two rhombuses \(ABCD\) and \(A'B'C'D'\) seem to be related by a \(180^{\circ}\) rotation or other transformations. Wait, actually, a reflection across the \(y\) - axis and then a \(180^{\circ}\) rotation: let's take a point \(A(10,6)\). Reflect across \(y\) - axis: \((-10,6)\), then rotate \(180^{\circ}\): \((10,-6)\). But in the graph, \(A'\) has coordinates \((6,10)\)? Wait, maybe my coordinate reading is wrong. Let's re - examine the graph. The \(x\) - axis and \(y\) - axis: the \(x\) - axis is horizontal (from left to right, values 2,4,6,8,10,12) and \(y\) - axis is vertical (from bottom to top, values 2,4,6,8,10,12). So point \(A\) is at \((10,6)\) (x = 10, y = 6), \(A'\) is at \((6,10)\)? Wait, no, maybe the axes are labeled differently. Wait, the \(x\) - axis is the horizontal axis with arrow pointing right (values 2,4,6,8,10,12) and \(y\) - axis is vertical with arrow pointing up (values 2,4,6,8,10,12). So the coordinates: for a point, \(x\) is the horizontal coordinate (left - right), \(y\) is vertical (bottom - top). So point \(A\): \(x = 10\), \(y = 6\); point \(A'\): \(x = 6\), \(y = 10\)? Wait, maybe it's a rotation. A \(90^{\circ}\) counterclockwise rotation about the origin transforms \((x,y)\) to \((-y,x)\). A \(90^{\circ}\) clockwise rotation transforms \((x,y)\) to \((y,-x)\). A \(180^{\circ}\) rotation transforms \((x,y)\) to \((-x,-y)\). A reflection across \(y\) - axis: \((-x,y)\), reflection across \(x\) - axis: \((x,-y)\).

• Option 2: \(90^{\circ}\) counterclockwise rotation then reflection across \(y\) - axis
• \(90^{\circ}\) counterclockwise rotation: \((x,y)\to(-y,x)\). Then reflection across \(y\) - axis: \((y,x)\). Let's take \(A(10,6)\): \(90^{\circ}\) counterclockwise: \((-6,10)\), then reflection across \(y\) - axis: \((6,10)\), which is the coordinates of \(A'\) (from the graph, \(A'\) seems to be at \((6,10)\)). So this transformation works.

• Option 3: Reflection across \(y\) - axis then \(90^{\circ}\) counterclockwise rotation
• Reflection across \(y\) - axis: \((x,y)\to(-x,y)\). Then \(90^{\circ}\) counterclockwise rotation: \((-y,-x)\). For \(A(10,6)\): reflection across \(y\) - axis: \((-10,6)\), then \(90^{\circ}\) counterclockwise: \((-6,-10)\), which is not \(A'\)'s coordinates. So this option is wrong.

• Option 4: \(90^{\circ}\) clockwise rotation then reflection across \(x\) - axis
• \(90^{\circ}\) clockwise rotation: \((x,y)\to(y,-x)\). Then reflection across \(x\) - axis: \((y,x)\). For \(A(10,6)\): \(90^{\circ}\) clockwise: \((6,-10)\), then reflection across \(x\) - axis: \((6,10)\), which is \(A'\)'s coordinates. So this transformation works.

• Option 5: Reflection across \(x\) - axis then \(90^{\circ}\) clockwise rotation
• Reflection across \(x\) - axis: \((x,y)\to(x,-y)\). Then \(90^{\circ}\) clockwise rotation: \((-y,x)\). For \(A(10,6)\): reflection across \(x\) - axis: \((10,-6)\), then \(90^{\circ}\) clockwise: \((6,10)\)? Wait, no: \(90^{\circ}\) clockwise rotation of \((10,-6)\) is \((-6,10)\)? Wait, no, the formula for \(90^{\circ}\) clockwise rotation is \((x,y)\to(y,-x)\). So \((10,-6)\) after \(90^{\circ}\) clockwise rotation is \((-6,-10)\), which is not \(A'\)'s coordinates. Wait, I think I made a mistake earlier. Let's re - calculate the \(90^{\circ}\) clockwise rotation: the standard matrix for \(90^{\circ}\) clockwise rotation is \(

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

\), so \(

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

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

=

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

\). So for \((x,y)=(10,6)\), \(90^{\circ}\) clockwise rotation is \((6,-10)\). Then reflection across \(x\) - axis: \((6,10)\), which is correct. For the \(90^{\circ}\) counterclockwise rotation, the matrix is \(

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

\), so \(

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

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

=

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

\). So for \((x,y)=(10,6)\), \(90^{\circ}\) counterclockwise rotation is \((-6,10)\), then reflection across \(y\) - axis: \(

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

$$\begin{pmatrix}-6\\10\end{pmatrix}$$

=

$$\begin{pmatrix}6\\10\end{pmatrix}$$

\), which is correct.

Wait, let's re - evaluate the options:

1. a reflection across the \(y\) - axis and then a \(180^{\circ}\) rotation about the origin:

• Let's take a point \(A(10,6)\). Reflect across \(y\) - axis: \((-10,6)\). Rotate \(180^{\circ}\) about the origin: \((10,-6)\), which is not \(A'\) (which is \((6,10)\)). So this option is incorrect.

1. a \(90^{\circ}\) counterclockwise rotation about the origin and then a reflection across the \(y\) - axis:

• \(A(10,6)\) rotated \(90^{\circ}\) counterclockwise: \((-6,10)\). Reflected across \(y\) - axis: \((6,10)\) (matches \(A'\)). So this is correct.

1. a reflection across the \(y\) - axis and then a \(90^{\circ}\) counterclockwise rotation about the origin:

• \(A(10,6)\) reflected across \(y\) - axis: \((-10,6)\). Rotated \(90^{\circ}\) counterclockwise: \((-6,-10)\) (does not match \(A'\)). Incorrect.

1. a \(90^{\circ}\) clockwise rotation about the origin and then a reflection across the \(x\) - axis:

• \(A(10,6)\) rotated \(90^{\circ}\) clockwise: \((6,-10)\). Reflected across \(x\) - axis: \((6,10)\) (matches \(A'\)). Correct.

1. a reflection across the \(x\) - axis and then a \(90^{\circ}\) clockwise rotation about the origin:

• \(A(10,6)\) reflected across \(x\) - axis: \((10,-6)\). Rotated \(90^{\circ}\) clockwise: \((-6,-10)\) (does not match \(A'\)). Incorrect.

Wait, maybe I misread the coordinates. Let's look at the graph again. The rhombus \(ABCD\) and \(A'B'C'D'\): the transformation from \(ABCD\) to \(A'B'C'D'\) seems to be a \(90^{\circ}\) rotation (either clockwise or counterclockwise) combined with a reflection, or a \(180^{\circ}\) rotation? Wait, another way: the two rhombuses are congruent, so we need to find the correct transformation sequences.

Wait, let's check the first option again: reflection across \(y\) - axis and then \(180^{\circ}\) rotation. The composition of \(R_y\) (reflection over \(y\)) and \(R_{180}\) (rotation \(180^{\circ}\)) is \(R_y\circ R_{180}(x,y)=R_y(-x,-y)=(x,-y)\), which is a reflection over \(x\) - axis. But the rhombus \(A'B'C'D'\) is not a reflection over \(x\) - axis of \(ABCD\). So option 1 is wrong.

Option 2: \(90^{\circ}\) counterclockwise rotation and then reflection over \(y\) - axis. The \(90^{\circ}\) counterclockwise rotation matrix is \(

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

\), so \((x,y)\to(-y,x)\). Then reflection over \(y\) - axis matrix is \(

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

\), so \((-y,x)\to(y,x)\). So the composition is \((x,y)\to(y,x)\), which is a reflection over the line \(y = x\). Let's check a point: if \(A\) is \((10,6)\), then \((y,x)=(6,10)\), which matches \(A'\) (assuming \(A'\) is \((6,10)\)). So this works.

Option 4: \(90^{\circ}\) clockwise rotation matrix is \(

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

\), so \((x,y)\to(y,-x)\). Then reflection over \(x\) - axis matrix is \(

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

\), so \((y,-x)\to(y,x)\). So the composition is also \((x,y)\to(y,x)\), same as option 2. So both option 2 and option 4 are correct? Wait, no, the option 4 says \(90^{\circ}\) clockwise rotation and then reflection across \(x\) - axis. So \((x,y)\to(y,-x)\to(y,x)\), which is correct. Option 2: \((x,y)\to(-y,x)\to(y,x)\), which is also correct. Wait, but let's check the other options.

Wait, the problem says "select all the correct answers". Let's re - evaluate each option:

1. a reflection across the \(y\) - axis and then a \(180^{\circ}\) rotation about the origin:

• As above, this gives \((x,-y)\), which is not the case. So wrong.

1. a \(90^{\circ}\) counterclockwise rotation about the origin and then a reflection across the \(y\) - axis:

• Correct, as shown.

1. a reflection across the \(y\) - axis and then a \(90^{\circ}\) counterclockwise rotation about the origin:

• Gives \((-y,-x)\), wrong.

1. a \(90^{\circ}\) clockwise rotation about the origin and then a reflection across the \(x\) - axis:

• Correct, as shown.

1. a reflection across the \(x\) - axis and then a \(90^{\circ}\) clockwise rotation about the origin:

• Gives \((-y,x)\)? Wait, no: reflection across \(x\) - axis: \((x,-y)\), then \(90^{\circ}\) clockwise rotation: \((-y,-x)\)? Wait, no, \(90^{\circ}\) clockwise rotation of \((x,-y)\) is \((-y,-x)\)? No, the formula for \(90^{\circ}\) clockwise rotation is \((x,y)\to(y,-x)\). So \((x,-y)\to(-y,-x)\), which is not the same as \(A'\). So option 5 is wrong.

Wait, maybe the first option is correct? Let's take a point \(C\) in \(ABCD\). Let's say \(C\) has coordinates \((6,2)\). Reflect across \(y\) - axis: \((-6,2)\), then rotate \(180^{\circ}\) about the origin: \((6,-2)\). But in \(A'B'C'D'\), \(C'\) has coordinates \((2,6)\)? Wait, I think I messed up the coordinate system. The \(x\) - axis and \(y\) - axis are labeled with \(x\) going from 2 to 12 (horizontal) and \(y\) going from 2 to 12 (vertical), but the origin is at the intersection of the axes (probably at (0,0), but the graph is shifted? No, the axes are drawn with \(x\) - axis (horizontal) and \(y\) - axis (vertical) intersecting at (0,0), and the grid lines are at \(x = 2,4,6,8,10,12\) and \(y = 2,4,6,8,10,12\). So a point with \(x = 6\) and \(y = 2\) is \((6,2)\).

Wait, maybe the correct options are:

• Option 2: \(90^{\circ}\) counterclockwise rotation about the origin and then a reflection across the \(y\) - axis.

• Option 4: \(90^{\circ}\) clockwise rotation about the origin and then a reflection across the \(x\) - axis.

• And maybe option 1? Wait, no, let's use the concept of congruence transformations (rotations, reflections, translations are congruence transformations).

Wait, another approach: the two rhombuses \(ABCD\) and \(A'B'C'D'\) are related by a \(90^{\circ}\) rotation (either clockwise or counterclockwise) combined with a reflection, or a \(180^{\circ}\) rotation? Wait, no, a \(180^{\circ}\) rotation would map \((x,y)\) to \((-x,-y)\), but the coordinates don't match. A \(90^{\circ}\) rotation (clockwise or counterclockwise) combined with a reflection gives the correct mapping.

So after re - analyzing, the correct options are:

• A \(90^{\circ}\) counterclockwise rotation about the origin and then a reflection across the \(y\) - axis.

• A \(90^{\circ}\) clockwise rotation about the origin and then a reflection across the \(x\) - axis.

Wait, but the first option: reflection across \(y\) - axis and