Rotating a Point Around a Fixed Point

Oct 13, 2025 by magicalceylontours.com 49 views

Introduction

In geometry, transformations are an essential concept that helps us understand how shapes change under various operations. One common type of transformation is a rotation, where a shape is rotated around a fixed point by a certain angle. In this article, we will explore a problem involving a triangle transformation, where we need to find the coordinates of a vertex after a 270° rotation.

Understanding the Problem

We are given a triangle with vertices at $R (1,1)$ , $S (-2,-4)$ , and $T (-3,-3)$ . The triangle is transformed according to the rule $R, 270^{\circ}$ . This means that the triangle is rotated 270° around the point $R (1,1)$ . Our goal is to find the coordinates of $S^{\prime}$ , the image of vertex $S$ after the rotation.

Rotating a Point Around a Fixed Point

To solve this problem, we need to understand how to rotate a point around a fixed point. Let's consider a point $P (x,y)$ and a fixed point $O (a,b)$ . We want to rotate $P$ around $O$ by an angle $\theta$ . The new coordinates of $P$ after the rotation can be found using the following formulas:

$x^{\prime} = a + (x-a)\cos\theta - (y-b)\sin\theta$

$y^{\prime} = b + (x-a)\sin\theta + (y-b)\cos\theta$

In our problem, the fixed point is $R (1,1)$ , and the angle of rotation is 270°. We can use the above formulas to find the coordinates of $S^{\prime}$ .

Finding the Coordinates of S'

Let's apply the rotation formulas to find the coordinates of $S^{\prime}$ . We have:

$x^{\prime} = 1 + (-2-1)\cos270^{\circ} - (-4-1)\sin270^{\circ}$

$y^{\prime} = 1 + (-2-1)\sin270^{\circ} + (-4-1)\cos270^{\circ}$

Since $\cos270^{\circ} = 0$ and $\sin270^{\circ} = -1$ , we can simplify the above equations:

$x^{\prime} = 1 + (-3)(0) - (-5)(-1)$

$x^{\prime} = 1 + 0 - 5$

$x^{\prime} = -4$

$y^{\prime} = 1 + (-3)(-1) + (-5)(0)$

$y^{\prime} = 1 + 3 + 0$

$y^{\prime} = 4$

Therefore, the coordinates of $S^{\prime}$ are $(-4,4)$ .

Conclusion

In this article, we solved a triangle transformation problem involving a 270° rotation around a fixed point. We used the rotation formulas to find the coordinates of $S^{\prime}$ , the image of vertex $S$ after the rotation. The final answer is $(-4,4)$ .

Frequently Asked Questions

What is the rule for rotating a point around a fixed point?
How do we find the coordinates of a point after a rotation?
What is the formula for rotating a point around a fixed point?

Step-by-Step Solution

Identify the fixed point and the angle of rotation.
Use the rotation formulas to find the new coordinates of the point.
Simplify the equations using the values of sine and cosine for the given angle.
Calculate the new coordinates of the point.

Key Concepts

Rotation
Fixed point
Angle of rotation
Rotation formulas
Sine and cosine values for 270°

Real-World Applications

Computer graphics
Game development
Architecture
Engineering

References

[1] Geometry textbook by [Author]
[2] Online resource for geometry and trigonometry by [Website]
[3] Math software and calculator by [Company]

Q&A: A Triangle Transformation Problem

Q: What is the rule for rotating a point around a fixed point?

A: The rule for rotating a point around a fixed point involves using the rotation formulas to find the new coordinates of the point. The formulas are:

$x^{\prime} = a + (x-a)\cos\theta - (y-b)\sin\theta$

$y^{\prime} = b + (x-a)\sin\theta + (y-b)\cos\theta$

where $(x,y)$ are the coordinates of the point, $(a,b)$ are the coordinates of the fixed point, and $\theta$ is the angle of rotation.

Q: How do we find the coordinates of a point after a rotation?

A: To find the coordinates of a point after a rotation, we need to use the rotation formulas and substitute the values of the fixed point and the angle of rotation. We can then simplify the equations using the values of sine and cosine for the given angle.

Q: What is the formula for rotating a point around a fixed point?

A: The formula for rotating a point around a fixed point is given by the rotation formulas:

$x^{\prime} = a + (x-a)\cos\theta - (y-b)\sin\theta$

$y^{\prime} = b + (x-a)\sin\theta + (y-b)\cos\theta$

Q: How do we determine the angle of rotation?

A: The angle of rotation is given in the problem statement. In this case, the angle of rotation is 270°.

Q: What is the significance of the fixed point in a rotation?

A: The fixed point is the point around which the rotation occurs. It remains stationary while the other points are rotated.

Q: Can we rotate a point by any angle?

A: Yes, we can rotate a point by any angle. However, the rotation formulas will change depending on the angle of rotation.

Q: How do we handle negative angles in a rotation?

A: Negative angles can be handled by adding 360° to the angle of rotation. For example, if the angle of rotation is -90°, we can add 360° to get 270°.

Q: Can we rotate a point by a non-integer angle?

A: Yes, we can rotate a point by a non-integer angle. However, the rotation formulas will involve trigonometric functions of the non-integer angle.

Q: How do we find the coordinates of a point after a rotation in 3D space?

A: To find the coordinates of a point after a rotation in 3D space, we need to use the rotation matrices and substitute the values of the fixed point and the angle of rotation. We can then multiply the rotation matrices to get the new coordinates of the point.

Q: Can we rotate a point by a combination of rotations?

A: Yes, we can rotate a point by a combination of rotations. However, the rotation formulas will involve the product of the rotation matrices.

Q: How do we handle multiple rotations in a single problem?

A: To handle multiple rotations in a single problem, we need to apply the rotation formulas in the correct order. We can then simplify the equations using the values of sine and cosine for the given angles.

Q: Can we rotate a point by a rotation around a line?

A: Yes, we can rotate a point by a rotation around a line. However, the rotation formulas will involve the angle of rotation and the distance from the point to the line.

Q: How do we find the coordinates of a point after a rotation around a line?

A: To find the coordinates of a point after a rotation around a line, we need to use the rotation formulas and substitute the values of the fixed point, the angle of rotation, and the distance from the point to the line. We can then simplify the equations using the values of sine and cosine for the given angle.

Q: Can we rotate a point by a rotation around a circle?

A: Yes, we can rotate a point by a rotation around a circle. However, the rotation formulas will involve the angle of rotation and the radius of the circle.

Q: How do we find the coordinates of a point after a rotation around a circle?

A: To find the coordinates of a point after a rotation around a circle, we need to use the rotation formulas and substitute the values of the fixed point, the angle of rotation, and the radius of the circle. We can then simplify the equations using the values of sine and cosine for the given angle.

Q: Can we rotate a point by a rotation around an ellipse?

A: Yes, we can rotate a point by a rotation around an ellipse. However, the rotation formulas will involve the angle of rotation and the semi-axes of the ellipse.

Q: How do we find the coordinates of a point after a rotation around an ellipse?

A: To find the coordinates of a point after a rotation around an ellipse, we need to use the rotation formulas and substitute the values of the fixed point, the angle of rotation, and the semi-axes of the ellipse. We can then simplify the equations using the values of sine and cosine for the given angle.

Q: Can we rotate a point by a rotation around a sphere?

A: Yes, we can rotate a point by a rotation around a sphere. However, the rotation formulas will involve the angle of rotation and the radius of the sphere.

Q: How do we find the coordinates of a point after a rotation around a sphere?

A: To find the coordinates of a point after a rotation around a sphere, we need to use the rotation formulas and substitute the values of the fixed point, the angle of rotation, and the radius of the sphere. We can then simplify the equations using the values of sine and cosine for the given angle.

Q: Can we rotate a point by a rotation around a torus?

A: Yes, we can rotate a point by a rotation around a torus. However, the rotation formulas will involve the angle of rotation and the major and minor radii of the torus.

Q: How do we find the coordinates of a point after a rotation around a torus?

A: To find the coordinates of a point after a rotation around a torus, we need to use the rotation formulas and substitute the values of the fixed point, the angle of rotation, and the major and minor radii of the torus. We can then simplify the equations using the values of sine and cosine for the given angle.

Q: Can we rotate a point by a rotation around a hyperboloid?

A: Yes, we can rotate a point by a rotation around a hyperboloid. However, the rotation formulas will involve the angle of rotation and the semi-axes of the hyperboloid.

Q: How do we find the coordinates of a point after a rotation around a hyperboloid?

A: To find the coordinates of a point after a rotation around a hyperboloid, we need to use the rotation formulas and substitute the values of the fixed point, the angle of rotation, and the semi-axes of the hyperboloid. We can then simplify the equations using the values of sine and cosine for the given angle.

Q: Can we rotate a point by a rotation around a paraboloid?

A: Yes, we can rotate a point by a rotation around a paraboloid. However, the rotation formulas will involve the angle of rotation and the semi-axes of the paraboloid.

Q: How do we find the coordinates of a point after a rotation around a paraboloid?

A: To find the coordinates of a point after a rotation around a paraboloid, we need to use the rotation formulas and substitute the values of the fixed point, the angle of rotation, and the semi-axes of the paraboloid. We can then simplify the equations using the values of sine and cosine for the given angle.

Q: Can we rotate a point by a rotation around a cone?

A: Yes, we can rotate a point by a rotation around a cone. However, the rotation formulas will involve the angle of rotation and the semi-axes of the cone.

Q: How do we find the coordinates of a point after a rotation around a cone?

A: To find the coordinates of a point after a rotation around a cone, we need to use the rotation formulas and substitute the values of the fixed point, the angle of rotation, and the semi-axes of the cone. We can then simplify the equations using the values of sine and cosine for the given angle.

Q: Can we rotate a point by a rotation around a cylinder?

A: Yes, we can rotate a point by a rotation around a cylinder. However, the rotation formulas will involve the angle of rotation and the radius of the cylinder.

Q: How do we find the coordinates of a point after a rotation around a cylinder?

A: To find the coordinates of a point after a rotation around a cylinder, we need to use the rotation formulas and substitute the values of the fixed point, the angle of rotation, and the radius of the cylinder. We can then simplify the equations using the values of sine and cosine for the given angle.

Q: Can we rotate a point by a rotation around a sphere with a hole?

A: Yes, we can rotate a point by a rotation around a sphere with a hole. However, the rotation formulas will involve the angle of rotation and the radius of the sphere.

Q: How do we find the coordinates of a point after a rotation around a sphere with a hole?

A: To find the coordinates of a point after a rotation around a sphere with a hole, we need to use the rotation formulas and substitute the values of the fixed point, the angle of rotation, and the radius of