What Are The Coordinates Of Each Point After Quadrilateral ABCD Is Rotated 270° Counterclockwise About The Origin? 6 4 2 ▬▬▬▬▬▬A DX -6-4-2 H 2 4 6 E2 FG 4 6 -5-4-3-2-1 1 2 3 4 5 A D. B' D. C D'

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Introduction

In geometry, a rotation is a transformation that turns a figure around a fixed point called the center of rotation. When a point is rotated 270° counterclockwise about the origin, its coordinates change. In this article, we will explore how to find the new coordinates of each point after a quadrilateral ABCD is rotated 270° counterclockwise about the origin.

Understanding Rotations

A rotation of 270° counterclockwise about the origin is equivalent to a rotation of 90° clockwise about the origin. This means that the new x-coordinate of a point will be the old y-coordinate, and the new y-coordinate will be the negative of the old x-coordinate.

Rotating Point A

Let's start by rotating point A, which has coordinates (6, 4). To rotate point A 270° counterclockwise about the origin, we need to swap its x and y coordinates and change the sign of the new x-coordinate.

x = 4
y = 6
new_x = -y
new_y = x
print(f"The new coordinates of point A are ({new_x}, {new_y})")

When we run this code, we get the following output:

The new coordinates of point A are (-6, 4)

Rotating Point B

Next, let's rotate point B, which has coordinates (4, 2). To rotate point B 270° counterclockwise about the origin, we need to swap its x and y coordinates and change the sign of the new x-coordinate.

x = 2
y = 4
new_x = -y
new_y = x
print(f"The new coordinates of point B are ({new_x}, {new_y})")

When we run this code, we get the following output:

The new coordinates of point B are (-4, 2)

Rotating Point C

Now, let's rotate point C, which has coordinates (2, 6). To rotate point C 270° counterclockwise about the origin, we need to swap its x and y coordinates and change the sign of the new x-coordinate.

x = 6
y = 2
new_x = -y
new_y = x
print(f"The new coordinates of point C are ({new_x}, {new_y})")

When we run this code, we get the following output:

The new coordinates of point C are (-2, 6)

Rotating Point D

Finally, let's rotate point D, which has coordinates (-6, -4). To rotate point D 270° counterclockwise about the origin, we need to swap its x and y coordinates and change the sign of the new x-coordinate.

x = -4
y = -6
new_x = -y
new_y = x
print(f"The new coordinates of point D are ({new_x}, {new_y})")

When we run this code, we get the following output:

The new coordinates of point D are (6, -4)

Conclusion

In this article, we explored how to find the new coordinates of each point after a quadrilateral ABCD is rotated 270° counterclockwise about the origin. We used Python code to demonstrate the rotation of each point and found the new coordinates of each point.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

  1. Identify the coordinates of each point in the quadrilateral ABCD.
  2. To rotate a point 270° counterclockwise about the origin, swap its x and y coordinates and change the sign of the new x-coordinate.
  3. Use Python code to demonstrate the rotation of each point and find the new coordinates of each point.

Example Use Case

Suppose we have a quadrilateral ABCD with coordinates A(6, 4), B(4, 2), C(2, 6), and D(-6, -4). We want to rotate this quadrilateral 270° counterclockwise about the origin. Using the steps outlined above, we can find the new coordinates of each point as follows:

  • Point A: (6, 4) → (-4, 6)
  • Point B: (4, 2) → (-2, 4)
  • Point C: (2, 6) → (-6, 2)
  • Point D: (-6, -4) → (4, -6)

Code Solution

Here is a Python code solution to the problem:

def rotate_point(x, y):
    new_x = -y
    new_y = x
    return new_x, new_y

A = (6, 4)
B = (4, 2)
C = (2, 6)
D = (-6, -4)



new_A = rotate_point(A[0], A[1])
new_B = rotate_point(B[0], B[1])
new_C = rotate_point(C[0], C[1])
new_D = rotate_point(D[0], D[1])



print(f"The new coordinates of point A are ({new_A[0]}, {new_A[1]})")
print(f"The new coordinates of point B are ({new_B[0]}, {new_B[1]})")
print(f"The new coordinates of point C are ({new_C[0]}, {new_C[1]})")
print(f"The new coordinates of point D are ({new_D[0]}, {new_D[1]})")

When we run this code, we get the following output:

The new coordinates of point A are (-4, 6)
The new coordinates of point B are (-2, 4)
The new coordinates of point C are (-6, 2)
The new coordinates of point D are (4, -6)

Final Answer

The new coordinates of each point after the quadrilateral ABCD is rotated 270° counterclockwise about the origin are:

  • Point A: (-4, 6)
  • Point B: (-2, 4)
  • Point C: (-6, 2)
  • Point D: (4, -6)

Introduction

In our previous article, we explored how to find the new coordinates of each point after a quadrilateral ABCD is rotated 270° counterclockwise about the origin. In this article, we will answer some frequently asked questions about rotating a quadrilateral 270° counterclockwise about the origin.

Q1: What is the formula for rotating a point 270° counterclockwise about the origin?

A1: The formula for rotating a point 270° counterclockwise about the origin is:

new_x = -y new_y = x

Where (x, y) are the original coordinates of the point.

Q2: How do I rotate a point 270° counterclockwise about the origin using Python?

A2: You can rotate a point 270° counterclockwise about the origin using Python as follows:

def rotate_point(x, y):
    new_x = -y
    new_y = x
    return new_x, new_y


x = 6
y = 4



new_x, new_y = rotate_point(x, y)



print(f"The new coordinates of the point are ({new_x}, {new_y})")

Q3: What is the difference between rotating a point 270° counterclockwise and rotating it 90° clockwise?

A3: Rotating a point 270° counterclockwise about the origin is equivalent to rotating it 90° clockwise about the origin. This means that the new x-coordinate of the point will be the old y-coordinate, and the new y-coordinate will be the negative of the old x-coordinate.

Q4: Can I rotate a point 270° counterclockwise about a point other than the origin?

A4: Yes, you can rotate a point 270° counterclockwise about a point other than the origin. To do this, you need to translate the point to the origin, rotate it 270° counterclockwise, and then translate it back to its original position.

Q5: How do I rotate a quadrilateral 270° counterclockwise about the origin using a graphing calculator?

A5: You can rotate a quadrilateral 270° counterclockwise about the origin using a graphing calculator as follows:

  1. Enter the coordinates of the quadrilateral into the calculator.
  2. Use the rotation function to rotate the quadrilateral 270° counterclockwise about the origin.
  3. Print the new coordinates of the quadrilateral.

Q6: Can I rotate a point 270° counterclockwise about the origin using a spreadsheet?

A6: Yes, you can rotate a point 270° counterclockwise about the origin using a spreadsheet as follows:

  1. Enter the coordinates of the point into a spreadsheet.
  2. Use the formula =-B2 to rotate the x-coordinate of the point 270° counterclockwise.
  3. Use the formula =A2 to rotate the y-coordinate of the point 270° counterclockwise.
  4. Print the new coordinates of the point.

Q7: How do I rotate a quadrilateral 270° counterclockwise about the origin using a programming language other than Python?

A7: You can rotate a quadrilateral 270° counterclockwise about the origin using a programming language other than Python as follows:

  1. Define the coordinates of the quadrilateral.
  2. Use the rotation formula to rotate each point of the quadrilateral 270° counterclockwise about the origin.
  3. Print the new coordinates of the quadrilateral.

Q8: Can I rotate a point 270° counterclockwise about the origin using a graphing software?

A8: Yes, you can rotate a point 270° counterclockwise about the origin using a graphing software as follows:

  1. Enter the coordinates of the point into the software.
  2. Use the rotation function to rotate the point 270° counterclockwise about the origin.
  3. Print the new coordinates of the point.

Q9: How do I rotate a quadrilateral 270° counterclockwise about the origin using a computer-aided design (CAD) software?

A9: You can rotate a quadrilateral 270° counterclockwise about the origin using a CAD software as follows:

  1. Enter the coordinates of the quadrilateral into the software.
  2. Use the rotation function to rotate the quadrilateral 270° counterclockwise about the origin.
  3. Print the new coordinates of the quadrilateral.

Q10: Can I rotate a point 270° counterclockwise about the origin using a mathematical software?

A10: Yes, you can rotate a point 270° counterclockwise about the origin using a mathematical software as follows:

  1. Enter the coordinates of the point into the software.
  2. Use the rotation formula to rotate the point 270° counterclockwise about the origin.
  3. Print the new coordinates of the point.

Conclusion

In this article, we answered some frequently asked questions about rotating a quadrilateral 270° counterclockwise about the origin. We provided step-by-step solutions to each question and demonstrated how to rotate a point 270° counterclockwise about the origin using various programming languages and software.