Plot A Point P(-3, 8) And Find Its Image: (i) P, When Reflected About The X-axis. (ii) P2 When Reflected About The Y-axis. (iii) P3 When Rotated Clockwise By 90° About The Origin. (iv) P4 When Rotated Anticlockwise By 90° About The Origin. (v) P5
Introduction
In geometry, transformations are used to change the position or orientation of a point or a shape. Reflection, rotation, and translation are some of the common transformations used in geometry. In this article, we will learn how to plot a point and find its image after different transformations.
Plotting a Point
To plot a point, we need to know its coordinates. The coordinates of a point are given in the form (x, y), where x is the x-coordinate and y is the y-coordinate.
Let's plot the point P(-3, 8).
Plotting Point P(-3, 8)
To plot the point P(-3, 8), we need to locate the point on the coordinate plane. The x-coordinate -3 indicates that the point is 3 units to the left of the y-axis, and the y-coordinate 8 indicates that the point is 8 units above the x-axis.
Reflection About the X-Axis
When a point is reflected about the x-axis, its y-coordinate changes sign. In other words, the y-coordinate becomes negative.
Let's find the image of point P(-3, 8) when reflected about the x-axis.
Image of P when Reflected About the X-Axis
The image of point P(-3, 8) when reflected about the x-axis is P'(-3, -8).
Reflection About the Y-Axis
When a point is reflected about the y-axis, its x-coordinate changes sign. In other words, the x-coordinate becomes negative.
Let's find the image of point P(-3, 8) when reflected about the y-axis.
Image of P2 when Reflected About the Y-Axis
The image of point P(-3, 8) when reflected about the y-axis is P2(3, 8).
Rotation Clockwise by 90°
When a point is rotated clockwise by 90° about the origin, its coordinates change as follows:
x' = y y' = -x
Let's find the image of point P(-3, 8) when rotated clockwise by 90° about the origin.
Image of P3 when Rotated Clockwise by 90°
The image of point P(-3, 8) when rotated clockwise by 90° about the origin is P3(8, 3).
Rotation Anticlockwise by 90°
When a point is rotated anticlockwise by 90° about the origin, its coordinates change as follows:
x' = -y y' = x
Let's find the image of point P(-3, 8) when rotated anticlockwise by 90° about the origin.
Image of P4 when Rotated Anticlockwise by 90°
The image of point P(-3, 8) when rotated anticlockwise by 90° about the origin is P4(-8, -3).
Translation
Translation is a transformation that moves a point from one location to another. The translation of a point is given by the formula:
(x', y') = (x + a, y + b)
where (x, y) is the original point, and (a, b) is the translation vector.
Let's find the image of point P(-3, 8) when translated by 2 units to the right and 3 units upwards.
Image of P5 when Translated
The image of point P(-3, 8) when translated by 2 units to the right and 3 units upwards is P5(-1, 11).
Conclusion
In this article, we learned how to plot a point and find its image after different transformations. We also learned how to reflect a point about the x-axis and y-axis, rotate a point clockwise and anticlockwise by 90°, and translate a point by a given vector.
Key Takeaways
- Reflection about the x-axis changes the sign of the y-coordinate.
- Reflection about the y-axis changes the sign of the x-coordinate.
- Rotation clockwise by 90° changes the coordinates as x' = y and y' = -x.
- Rotation anticlockwise by 90° changes the coordinates as x' = -y and y' = x.
- Translation moves a point from one location to another by a given vector.
Practice Problems
- Plot the point P(4, -2) and find its image when reflected about the x-axis.
- Plot the point P(-5, 3) and find its image when reflected about the y-axis.
- Plot the point P(2, 6) and find its image when rotated clockwise by 90° about the origin.
- Plot the point P(-1, -4) and find its image when rotated anticlockwise by 90° about the origin.
- Plot the point P(3, 1) and find its image when translated by 2 units to the right and 3 units upwards.
Geometric Transformations: Q&A ================================
Introduction
In our previous article, we learned about geometric transformations, including reflection, rotation, and translation. In this article, we will answer some frequently asked questions about geometric transformations.
Q: What is the difference between reflection and rotation?
A: Reflection is a transformation that changes the sign of the x-coordinate or y-coordinate of a point, while rotation is a transformation that changes the coordinates of a point in a circular motion.
Q: How do I reflect a point about the x-axis?
A: To reflect a point about the x-axis, you need to change the sign of the y-coordinate. For example, if the point is (x, y), its reflection about the x-axis is (x, -y).
Q: How do I reflect a point about the y-axis?
A: To reflect a point about the y-axis, you need to change the sign of the x-coordinate. For example, if the point is (x, y), its reflection about the y-axis is (-x, y).
Q: How do I rotate a point clockwise by 90°?
A: To rotate a point clockwise by 90°, you need to change the coordinates as follows: x' = y and y' = -x.
Q: How do I rotate a point anticlockwise by 90°?
A: To rotate a point anticlockwise by 90°, you need to change the coordinates as follows: x' = -y and y' = x.
Q: What is the formula for translation?
A: The formula for translation is (x', y') = (x + a, y + b), where (x, y) is the original point and (a, b) is the translation vector.
Q: How do I find the image of a point after translation?
A: To find the image of a point after translation, you need to add the translation vector to the coordinates of the original point.
Q: Can I combine multiple transformations?
A: Yes, you can combine multiple transformations to create a single transformation. For example, you can reflect a point about the x-axis and then rotate it clockwise by 90°.
Q: What are some real-world applications of geometric transformations?
A: Geometric transformations have many real-world applications, including computer graphics, game development, and architecture. They are also used in science and engineering to model and analyze complex systems.
Q: How can I practice geometric transformations?
A: You can practice geometric transformations by working on problems and exercises, such as plotting points and finding their images after different transformations. You can also use online resources and software to visualize and explore geometric transformations.
Conclusion
In this article, we answered some frequently asked questions about geometric transformations. We hope that this article has helped you to better understand these important concepts and to develop your skills in working with geometric transformations.
Key Takeaways
- Reflection changes the sign of the x-coordinate or y-coordinate of a point.
- Rotation changes the coordinates of a point in a circular motion.
- Translation moves a point from one location to another by a given vector.
- You can combine multiple transformations to create a single transformation.
- Geometric transformations have many real-world applications.
Practice Problems
- Reflect the point (3, 4) about the x-axis.
- Reflect the point (-2, 5) about the y-axis.
- Rotate the point (1, 2) clockwise by 90°.
- Rotate the point (-3, 4) anticlockwise by 90°.
- Translate the point (2, 3) by 2 units to the right and 3 units upwards.