Demonstrating The Properties Of Rotations, If A Line Segment With Endpoints { (0,-3)$}$ And { (0,-7)$}$ Is Rotated ${ 90^{\circ}\$} Clockwise, What Is An Endpoint Of This Rotated Segment?A. { (0,3)$}$B.

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Introduction

Rotations are an essential concept in geometry and trigonometry, and understanding their properties is crucial for solving various mathematical problems. In this article, we will demonstrate the properties of rotations by considering a line segment with endpoints (0,−3)(0,-3) and (0,−7)(0,-7) that is rotated 90∘90^{\circ} clockwise. We will explore the effects of this rotation on the line segment and determine the endpoint of the rotated segment.

Understanding Rotations

A rotation is a transformation that turns a figure around a fixed point called the center of rotation. The amount of rotation is measured in degrees, and a positive rotation is clockwise, while a negative rotation is counterclockwise. In this case, we are rotating the line segment 90∘90^{\circ} clockwise, which means that the segment will be turned to the right by 90∘90^{\circ}.

Visualizing the Rotation

To visualize the rotation, let's consider the line segment with endpoints (0,−3)(0,-3) and (0,−7)(0,-7). We can draw a diagram to represent this segment and its rotation.

  +---------------+
  |               |
  |  (0, -7)     |
  |  (0, -3)     |
  |               |
  +---------------+

In this diagram, the line segment is represented by the two points (0,−3)(0,-3) and (0,−7)(0,-7). When we rotate this segment 90∘90^{\circ} clockwise, the new endpoint will be located at a different position.

Determining the Endpoint

To determine the endpoint of the rotated segment, we need to consider the effect of the rotation on the coordinates of the original endpoint. When a point is rotated 90∘90^{\circ} clockwise, its new coordinates can be found using the following formulas:

x′=yx' = y y′=−xy' = -x

where (x,y)(x,y) are the original coordinates and (x′,y′)(x',y') are the new coordinates.

Applying the Formulas

Let's apply these formulas to the original endpoint (0,−3)(0,-3). We can substitute x=0x=0 and y=−3y=-3 into the formulas to find the new coordinates:

x′=−3x' = -3 y′=−0y' = -0

Therefore, the new endpoint of the rotated segment is (0,3)(0,3).

Conclusion

In this article, we demonstrated the properties of rotations by considering a line segment with endpoints (0,−3)(0,-3) and (0,−7)(0,-7) that is rotated 90∘90^{\circ} clockwise. We explored the effects of this rotation on the line segment and determined the endpoint of the rotated segment using the formulas for rotating a point 90∘90^{\circ} clockwise. The new endpoint of the rotated segment is (0,3)(0,3).

Frequently Asked Questions

Q: What is the effect of rotating a point 90∘90^{\circ} clockwise?

A: When a point is rotated 90∘90^{\circ} clockwise, its new coordinates can be found using the formulas x′=yx' = y and y′=−xy' = -x.

Q: How do I determine the endpoint of a rotated segment?

A: To determine the endpoint of a rotated segment, you can use the formulas for rotating a point 90∘90^{\circ} clockwise and substitute the original coordinates into the formulas.

Q: What is the center of rotation?

A: The center of rotation is the fixed point around which the rotation takes place.

References

  • [1] Geometry and Trigonometry, by Michael Artin
  • [2] Rotations in Geometry, by David A. Brannan

Further Reading

  • Rotations in 2D Geometry
  • Rotations in 3D Geometry
  • Applications of Rotations in Geometry and Trigonometry

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Introduction

Rotations are an essential concept in geometry and trigonometry, and understanding their properties is crucial for solving various mathematical problems. In this article, we will answer some frequently asked questions about rotations in geometry, covering topics such as the effect of rotating a point, determining the endpoint of a rotated segment, and the center of rotation.

Q&A

Q: What is the effect of rotating a point 90∘90^{\circ} clockwise?

A: When a point is rotated 90∘90^{\circ} clockwise, its new coordinates can be found using the formulas x′=yx' = y and y′=−xy' = -x. This means that the x-coordinate becomes the y-coordinate, and the y-coordinate becomes the negative of the x-coordinate.

Q: How do I determine the endpoint of a rotated segment?

A: To determine the endpoint of a rotated segment, you can use the formulas for rotating a point 90∘90^{\circ} clockwise and substitute the original coordinates into the formulas. For example, if the original endpoint is (x,y)(x,y), the new endpoint will be (y,−x)(y,-x).

Q: What is the center of rotation?

A: The center of rotation is the fixed point around which the rotation takes place. It is the point that remains unchanged during the rotation.

Q: Can a point be rotated by an angle other than 90∘90^{\circ}?

A: Yes, a point can be rotated by any angle, not just 90∘90^{\circ}. The formulas for rotating a point by an angle θ\theta are:

x′=xcos⁡θ−ysin⁡θx' = x \cos \theta - y \sin \theta y′=xsin⁡θ+ycos⁡θy' = x \sin \theta + y \cos \theta

Q: How do I determine the angle of rotation?

A: To determine the angle of rotation, you can use the fact that the rotation is a rigid motion, which means that the distance between the original point and the rotated point remains unchanged. You can use the distance formula to find the angle of rotation.

Q: Can a line segment be rotated by a negative angle?

A: Yes, a line segment can be rotated by a negative angle. A negative angle of rotation is equivalent to a positive angle of rotation in the opposite direction.

Q: What is the effect of rotating a line segment by 180∘180^{\circ}?

A: When a line segment is rotated by 180∘180^{\circ}, the new endpoint will be the same distance from the center of rotation as the original endpoint, but in the opposite direction.

Examples

Example 1: Rotating a Point 90∘90^{\circ} Clockwise

Suppose we want to rotate the point (2,3)(2,3) 90∘90^{\circ} clockwise. Using the formulas for rotating a point 90∘90^{\circ} clockwise, we get:

x′=3x' = 3 y′=−2y' = -2

Therefore, the new endpoint of the rotated segment is (3,−2)(3,-2).

Example 2: Rotating a Line Segment by 180∘180^{\circ}

Suppose we want to rotate the line segment with endpoints (0,0)(0,0) and (2,3)(2,3) by 180∘180^{\circ}. Using the formulas for rotating a line segment by 180∘180^{\circ}, we get:

x′=−2x' = -2 y′=−3y' = -3

Therefore, the new endpoint of the rotated segment is (−2,−3)(-2,-3).

Conclusion

In this article, we answered some frequently asked questions about rotations in geometry, covering topics such as the effect of rotating a point, determining the endpoint of a rotated segment, and the center of rotation. We also provided examples to illustrate the concepts.

Frequently Asked Questions

Q: What is the effect of rotating a point 90∘90^{\circ} clockwise?

A: When a point is rotated 90∘90^{\circ} clockwise, its new coordinates can be found using the formulas x′=yx' = y and y′=−xy' = -x.

Q: How do I determine the endpoint of a rotated segment?

A: To determine the endpoint of a rotated segment, you can use the formulas for rotating a point 90∘90^{\circ} clockwise and substitute the original coordinates into the formulas.

Q: What is the center of rotation?

A: The center of rotation is the fixed point around which the rotation takes place.

References

  • [1] Geometry and Trigonometry, by Michael Artin
  • [2] Rotations in Geometry, by David A. Brannan

Further Reading

  • Rotations in 2D Geometry
  • Rotations in 3D Geometry
  • Applications of Rotations in Geometry and Trigonometry