How to Find the Reflection of a Point in Coordinate Geometry
The concept of reflection in maths plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Reflection in Maths?
A reflection in maths is a transformation that “flips” a point, line, or shape over a specific line known as the line of reflection—creating a mirror image. You’ll find this concept applied in areas such as geometry, coordinate geometry, and symmetry problems. Understanding reflection in maths helps you solve questions on coordinate geometry, spot symmetry in nature, and interpret patterns in real life or art.
Key Formula for Reflection in Maths
Here’s the standard formula: If a point \( (x, y) \) is reflected, the coordinates of its image change depending on the axis or line of reflection.
| Reflection Line | New Coordinates |
|---|---|
| x-axis | (x, -y) |
| y-axis | (-x, y) |
| y = x | (y, x) |
| y = -x | (-y, -x) |
| Origin (0, 0) | (-x, -y) |
Memorizing these formulas will make you fast and accurate when solving reflection questions in competitive exams.
Cross-Disciplinary Usage
Reflection in maths is not only useful in Maths but also plays an important role in Physics (e.g., mirror reflections, optics), Computer Science (graphics, game development), and daily logical reasoning. Students preparing for JEE, Olympiads, or even CBSE boards will see its relevance in many chapters and questions.
Step-by-Step Illustration
- Given point: (5, 3). Required: Find its reflection across the x-axis.
The formula for x-axis reflection is (x, -y). - Apply the formula:
Image = (5, -3) - Final Answer:
The reflection of (5, 3) across the x-axis is (5, -3).
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for reflection in coordinate geometry: Notice the axis or line, and simply switch the sign or swap the coordinates as per the formula. For y = x, just swap x and y. For the x-axis or y-axis, just change the relevant sign. This helps you work super-fast in MCQs.
Example Trick: For (a, b), reflection over the y-axis is always (–a, b). No matter how big the numbers, just put a minus sign before x!
- Start with (7, 12). Required: Reflect across y-axis.
- Swap the sign of x: (-7, 12)
- Done! This works instantly for MCQs and one-mark exam questions.
Tricks like these aren’t just smart—they’re practical for CBSE boards, JEE Mains, and Olympiads. Vedantu’s live classes offer many such quick tips for building your confidence.
Try These Yourself
- Find the image of (9, –8) after reflection over the y-axis.
- What is the reflection of (–2, 5) across the line y = x?
- If a point P (x, y) is its own reflection across the x-axis, what must y be?
- Reflect (4, –1) about the origin.
Frequent Errors and Misunderstandings
- Swapping both signs regardless of axis (e.g., using (–x, –y) when reflecting only over x- or y-axis).
- Mixing up reflection with rotation or translation in transformation problems.
- Forgetting to change the correct coordinate (e.g., flipping y instead of x when reflecting over y-axis).
Relation to Other Concepts
The idea of reflection in maths connects closely with reflection symmetry, translations in maths, and congruence (congruence of triangles). Mastering reflections helps you ace questions on symmetry, transformations, and understanding graphs in coordinate geometry. It is also helpful in comparing with properties of triangle or polygons.
Classroom Tip
A quick way to remember reflection in maths is to visualize folding the paper along the line of reflection. Each point and its image remain at the same distance from this line but on opposite sides. Vedantu’s teachers often use colored dots or graph paper in their classes to help students “see” reflections instantly.
We explored reflection in maths—from definition, formula, examples, mistakes, and its relation to other geometry concepts. Continue practicing with Vedantu to become confident in quickly solving exam questions using this transformation. Check out more on line of symmetry and polygons for deeper mastery!
FAQs on Reflection (in Maths): Concept, Formula & Solved Examples
1. What is reflection in mathematics?
In mathematics, reflection is a transformation that flips a shape or point over a specific line, called the line of reflection, creating a mirror image. The reflected image maintains the same size and shape as the original, but its orientation is reversed.
2. What is the reflection formula for a point across the x-axis?
To reflect a point (x, y) across the x-axis, the x-coordinate remains the same, while the y-coordinate changes its sign. The reflection formula is: (x, -y)
3. How do you reflect a point across the y-axis?
Reflecting a point (x, y) across the y-axis involves changing the sign of the x-coordinate while keeping the y-coordinate unchanged. The reflection formula is: (-x, y)
4. How do you reflect a point across the line y = x?
To reflect a point (x, y) across the line y = x, simply swap the x and y coordinates. The reflection formula is: (y, x)
5. How do you reflect a point across the line y = -x?
Reflecting a point (x, y) across the line y = -x involves swapping the x and y coordinates and then negating both. The reflection formula is: (-y, -x)
6. How do you reflect a shape in coordinate geometry?
To reflect a shape, reflect each of its vertices using the appropriate reflection formula (depending on the line of reflection). Then, connect the reflected vertices to form the reflected shape. The reflected shape will be congruent to the original.
7. What is the difference between reflection and rotation?
Reflection flips a shape across a line, reversing its orientation. Rotation turns a shape around a point, maintaining its orientation. Both are transformations that preserve size and shape (isometries).
8. What is the difference between reflection and translation?
Reflection flips a shape across a line, while translation slides a shape to a new position without changing its orientation. Reflection reverses orientation, translation does not.
9. What are some real-life examples of reflection?
Real-life examples of reflection include:
- Your image in a mirror
- The reflection of the sun in water
- Patterns in nature exhibiting reflectional symmetry
10. What are the properties of reflection?
Key properties of reflection include:
- The line of reflection is the perpendicular bisector of the segment connecting a point and its reflection.
- The reflected image is congruent to the original shape.
- Reflection preserves distances between points.
- Reflection reverses the orientation of the shape.
11. How is reflection used in proving triangle congruence?
Reflection can be used as a transformation to show that two triangles are congruent. If one triangle can be mapped onto another through a reflection (or a sequence of reflections and other isometries), then the triangles are congruent.
12. What is the effect of successive reflections over parallel lines?
Successive reflections over parallel lines result in a translation. The distance of the translation is twice the distance between the parallel lines.
