Question

rotate \\(\triangle xyz\\) \\(270^\circ\\) about the origin, \\((0,0)\\)
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\\(\circ\\) a
\\(\circ\\) b
\\(\circ\\) c

Explanation:

To solve the problem of rotating \(\triangle XYZ\) \(270^\circ\) about the origin \((0,0)\), we use the rotation rule for a \(270^\circ\) counterclockwise (or \(90^\circ\) clockwise) rotation about the origin: \((x,y)\to(y, -x)\).

Step 1: Identify Coordinates of Vertices

First, we need the coordinates of \(X\), \(Y\), and \(Z\) from the original triangle. Let's assume (from typical grid problems) the coordinates are:

• \(X=(a,b)\)
• \(Y=(c,d)\)
• \(Z=(e,f)\)

Step 2: Apply the \(270^\circ\) Rotation Rule

For a \(270^\circ\) counterclockwise rotation about the origin, the transformation is:
\[
(x,y) \xrightarrow{270^\circ \text{ rotation}} (y, -x)
\]

Let’s apply this to each vertex:

• For \(X=(a,b)\): New coordinates \(X'=(b, -a)\)
• For \(Y=(c,d)\): New coordinates \(Y'=(d, -c)\)
• For \(Z=(e,f)\): New coordinates \(Z'=(f, -e)\)

Step 3: Match to the Correct Graph

After applying the rotation, we compare the new coordinates of \(X'\), \(Y'\), and \(Z'\) to the options (A, B, C, etc.). The correct graph will have vertices at \((b, -a)\), \((d, -c)\), and \((f, -e)\).

Example (Assuming Original Coordinates)

Suppose the original coordinates are:

• \(X=(2, 3)\), \(Y=(5, 3)\), \(Z=(2, 1)\)

Applying \(270^\circ\) rotation:

• \(X'=(3, -2)\)
• \(Y'=(3, -5)\)
• \(Z'=(1, -2)\)

We then check which graph has these new coordinates.

Final Answer

The correct option (e.g., if the rotated triangle matches Option C) is:

\(\boldsymbol{\text{C}}\) (or the specific option with the rotated vertices).

(Note: The exact answer depends on the original coordinates of \(X\), \(Y\), \(Z\) from the grid. The key is applying the \(270^\circ\) rotation rule \((x,y)\to(y, -x)\) to each vertex and matching to the graph.)

Answer:

To solve the problem of rotating \(\triangle XYZ\) \(270^\circ\) about the origin \((0,0)\), we use the rotation rule for a \(270^\circ\) counterclockwise (or \(90^\circ\) clockwise) rotation about the origin: \((x,y)\to(y, -x)\).

Step 1: Identify Coordinates of Vertices

First, we need the coordinates of \(X\), \(Y\), and \(Z\) from the original triangle. Let's assume (from typical grid problems) the coordinates are:

• \(X=(a,b)\)
• \(Y=(c,d)\)
• \(Z=(e,f)\)

Step 2: Apply the \(270^\circ\) Rotation Rule

For a \(270^\circ\) counterclockwise rotation about the origin, the transformation is:
\[
(x,y) \xrightarrow{270^\circ \text{ rotation}} (y, -x)
\]

Let’s apply this to each vertex:

• For \(X=(a,b)\): New coordinates \(X'=(b, -a)\)
• For \(Y=(c,d)\): New coordinates \(Y'=(d, -c)\)
• For \(Z=(e,f)\): New coordinates \(Z'=(f, -e)\)

Step 3: Match to the Correct Graph

After applying the rotation, we compare the new coordinates of \(X'\), \(Y'\), and \(Z'\) to the options (A, B, C, etc.). The correct graph will have vertices at \((b, -a)\), \((d, -c)\), and \((f, -e)\).

Example (Assuming Original Coordinates)

Suppose the original coordinates are:

• \(X=(2, 3)\), \(Y=(5, 3)\), \(Z=(2, 1)\)

Applying \(270^\circ\) rotation:

• \(X'=(3, -2)\)
• \(Y'=(3, -5)\)
• \(Z'=(1, -2)\)

We then check which graph has these new coordinates.

Final Answer

The correct option (e.g., if the rotated triangle matches Option C) is:

\(\boldsymbol{\text{C}}\) (or the specific option with the rotated vertices).

(Note: The exact answer depends on the original coordinates of \(X\), \(Y\), \(Z\) from the grid. The key is applying the \(270^\circ\) rotation rule \((x,y)\to(y, -x)\) to each vertex and matching to the graph.)