Question

1-85b. use the online etool from the last question to help you with this question: abcd is rotated 90° clockwise (↻) about the origin to form a′b′c′d′, what are the coordinates of the vertices of a′b′c′d′? note: write your answers as points: (x,y) with no extra spaces. a′ is at blank b′ is at blank c′ is at blank d′ is at blank

Explanation:

To solve the problem of rotating quadrilateral \(ABCD\) \(90^\circ\) clockwise about the origin, we need the coordinates of the original vertices \(A\), \(B\), \(C\), and \(D\). Since they are not provided in the question, we'll assume a common example (e.g., \(A(-2, 1)\), \(B(-1, 3)\), \(C(2, 2)\), \(D(1, 0)\)) to demonstrate the rotation rule.

Rotation Rule for \(90^\circ\) Clockwise About the Origin:

For a point \((x, y)\), the new coordinates after a \(90^\circ\) clockwise rotation are \((y, -x)\).

Step 1: Rotate \(A(x, y)\)

Let \(A = (-2, 1)\). Applying the rule:
New \(x\)-coordinate: \(y = 1\)
New \(y\)-coordinate: \(-x = -(-2) = 2\)
Thus, \(A' = (1, 2)\).

Step 2: Rotate \(B(x, y)\)

Let \(B = (-1, 3)\). Applying the rule:
New \(x\)-coordinate: \(y = 3\)
New \(y\)-coordinate: \(-x = -(-1) = 1\)
Thus, \(B' = (3, 1)\).

Step 3: Rotate \(C(x, y)\)

Let \(C = (2, 2)\). Applying the rule:
New \(x\)-coordinate: \(y = 2\)
New \(y\)-coordinate: \(-x = -2\)
Thus, \(C' = (2, -2)\).

Step 4: Rotate \(D(x, y)\)

Let \(D = (1, 0)\). Applying the rule:
New \(x\)-coordinate: \(y = 0\)
New \(y\)-coordinate: \(-x = -1\)
Thus, \(D' = (0, -1)\).

Final Answer (Using the Assumed Coordinates):

• \(A'\) is at \(\boldsymbol{(1, 2)}\)
• \(B'\) is at \(\boldsymbol{(3, 1)}\)
• \(C'\) is at \(\boldsymbol{(2, -2)}\)
• \(D'\) is at \(\boldsymbol{(0, -1)}\)

Note: If the original coordinates of \(ABCD\) differ, substitute them into the rotation rule \((x, y) \to (y, -x)\) to find the correct coordinates for \(A'B'C'D'\).

Answer:

To solve the problem of rotating quadrilateral \(ABCD\) \(90^\circ\) clockwise about the origin, we need the coordinates of the original vertices \(A\), \(B\), \(C\), and \(D\). Since they are not provided in the question, we'll assume a common example (e.g., \(A(-2, 1)\), \(B(-1, 3)\), \(C(2, 2)\), \(D(1, 0)\)) to demonstrate the rotation rule.

Rotation Rule for \(90^\circ\) Clockwise About the Origin:

For a point \((x, y)\), the new coordinates after a \(90^\circ\) clockwise rotation are \((y, -x)\).

Step 1: Rotate \(A(x, y)\)

Let \(A = (-2, 1)\). Applying the rule:
New \(x\)-coordinate: \(y = 1\)
New \(y\)-coordinate: \(-x = -(-2) = 2\)
Thus, \(A' = (1, 2)\).

Step 2: Rotate \(B(x, y)\)

Let \(B = (-1, 3)\). Applying the rule:
New \(x\)-coordinate: \(y = 3\)
New \(y\)-coordinate: \(-x = -(-1) = 1\)
Thus, \(B' = (3, 1)\).

Step 3: Rotate \(C(x, y)\)

Let \(C = (2, 2)\). Applying the rule:
New \(x\)-coordinate: \(y = 2\)
New \(y\)-coordinate: \(-x = -2\)
Thus, \(C' = (2, -2)\).

Step 4: Rotate \(D(x, y)\)

Let \(D = (1, 0)\). Applying the rule:
New \(x\)-coordinate: \(y = 0\)
New \(y\)-coordinate: \(-x = -1\)
Thus, \(D' = (0, -1)\).

Final Answer (Using the Assumed Coordinates):

• \(A'\) is at \(\boldsymbol{(1, 2)}\)
• \(B'\) is at \(\boldsymbol{(3, 1)}\)
• \(C'\) is at \(\boldsymbol{(2, -2)}\)
• \(D'\) is at \(\boldsymbol{(0, -1)}\)

Note: If the original coordinates of \(ABCD\) differ, substitute them into the rotation rule \((x, y) \to (y, -x)\) to find the correct coordinates for \(A'B'C'D'\).