Flipping is a term used in geometry to describe a reflection. A mirror image of a shape is called a reflection. The line of reflection is a line along which an image reflects. Reflection of point using graph paper is described as a figure that is built around a single point. It is also known as a point of reflection or its centre. Reflection on graph paper (on the cartesian place) is for \(X\) and \(Y\) axis.

If a figure is stated to mirror another figure, then each point in the first figure is equidistant from the corresponding point in the second figure. The reflected picture should have the same shape and size as the original, but it should face differently. The translation may occur because of changes in position during reflection. In this article, we will provide detailed information on the reflection of point using graph paper. Scroll down to learn more!

Learn About the Phenomenon of Reflection Here

Reflection Definition

Mirror reflection is similar to line symmetry. When we position an object near a mirror, we see a reflection of the object in the mirror. The size and shape of the reflected picture are identical to those of the original item. Consider the following figure:

Follow the methods below to determine a figure’s reflection symmetry:
1. Draw the mirror line.
2. Examine the figure to see if the symmetry line divides it into two identical sections.
When an object and its reflected image are reflected on a mirror, the object and its reflected image are identical, and the mirror line is the line of reflection symmetry. Line symmetry is also referred to as reflection symmetry or mirror symmetry since it is comparable to reflection symmetry.
Mirror reflection is when the mirror is placed in the vertical direction.

Mirror reflection is when the mirror is placed in the horizontal direction.

Important Points on Reflection

1. In geometry, flipping an image is known as a reflection. As a result, the final image will reflect the original structure.
2. The only difference is that the generated image is in the opposite direction. The size and shape, however, remain the same.
3. Because the position has altered because of the transformation, there are possibilities for translation as well.
4. Both figures (before and after reflection) are equidistant from all places on their respective surfaces.
5. The line of reflection refers to a single line that aids in the reflection of an object.
Note: Any object’s reflected image is identical to the object itself. Reflection symmetry is a concept utilised in the design of astronomical telescopes.

Reflection of Point in a Line

If you look at the image of a point \(A\) in the mirror \(MR,\) as shown in the figure, you will find it at \(A’.\) Let \(AA’\) intersect \(MR\) at \(O.\) We find that
1. \(OA=OA’.\) The distance of a point from the mirror is the same as the distance of its image.
2. \(MR \bot AA’.\) The mirror line is perpendicular bisector to the line joining the point and its image.

Remember: The mirror line is the perpendicular bisector of the line segment joining a point and reflection (image).

Note: The image of any point on the mirror line will be the point itself.

Reflections in the Coordinate Plane

A reflection is a metamorphosis that depicts a figure being flipped. A point, a line, or a plane can all be used to reflect figures. The picture of a figure reflected in a line or a point is consistent with the pre-image. Every point in a figure is mapped to an image by a reflection across a set line. The line of reflection refers to a fixed-line. Reflection in the coordinate plane is based on whether the reflection is over \(X\)-axis, \(Y\)-axis and in the origin \(\left( {0,\,0} \right).\)

Reflection Over X-axis

The \(x\)-coordinate remains the same when a point is reflected across the \(x\)-axis, while the \(y\)-coordinate is turned into the opposite (its sign is changed).
When graphing, if you forget the rules for reflections, fold the paper along the \(x\)-axis (the line of reflection) to see where the new figure will be.

The point \(\left( {x,\,y} \right)\) reflected across the \(x\)-axis is \(\left( {x,\, – y} \right).\)

Reflection Over Y-axis

The \(y\)-coordinate remains the same when a point is reflected across the \(y\)-axis, while the \(x\)-coordinate is changed into the opposite (its sign is changed).

The point \(\left( {x,\,y} \right)\) is the reflection of the point \(\left( {x,\,y} \right)\) across the \(y\)-axis \(\left( { – x,\,y} \right).\)

Reflection Over Y=X

The \(x\)-coordinate and \(y\)-coordinate swap places when a point is reflected across the line \(y = x.\) The \(x\)-coordinates and \(y\)-coordinates change places and are negated if you reflect over the line \(y = \, – x\) (the signs are changed). The point \(\left( {x,\,y} \right)\) reflected across the line \(y = x\) is the point \(\left( {y,\,x} \right).\)
The point \(\left( {x,\,y} \right)\) reflected across the line \(y = \, – x\) is the point \(\left( { – y,\, – x} \right).\)

Reflection in a Point

When a figure is built around a single point known as the point of reflection or the figure’s centre, it becomes a reflection point. On the other side of every point in the diagram, another point is discovered precisely opposite it. The size and shape of the figure do not change when it is reflected.

Reflection in Origin (0, 0)

After a point reflected in origin, triangle \(A’B’C’\) is the image of triangle \(ABC.\) Consider a straight-line joining point \(A\) and \(A’,\) with the origin at the segment’s midway.
Both the \(x\)-coordinate and the \(y\)-coordinate are negated when you reflect a point in the origin (their signs are changed). The point \(\left( { – x,\, – y} \right)\) is the image of the point \(\left( {x,\,y} \right)\) when reflected in origin \(\left( {0,\,0} \right).\)

Solved Examples on Reflection of Point Using Graph Paper

Q.1. Write down the coordinates of the images of the points. \(P\left( {7,\,3} \right),\,Q\left( {8,\, – 4} \right),\,R\left( { – 3,\,6} \right)\) and \(S\left( { – 4,\, – 2} \right)\) when reflected in the \(x\)-axis.
Ans: When a point is reflected in the \(x\)-axis, the \(x\)-coordinate remains the same while the y coordinate changes its sign.
\(P\left( {7,3} \right) \to P’\left( {7,\, – 3} \right)\) under reflection in the \(x\)-axis.
\(Q\left( {8,\, – 4} \right) \to Q’\left( {8,4} \right)\) under reflection in the \(x\)-axis.
\(R\left( { – 3,\,6} \right) \to R’\left( { – 3,\, – 6} \right)\) under reflection in the \(x\)-axis.
\(S\left( { – 4,\, – 2} \right) \to S’\left( { – 4,2} \right)\) under reflection in the \(x\)-axis.

Q.2. Write down the coordinates of the images of the points \(D\left( {4,5} \right),\,E\left( {7,\, – 3} \right),\,F\left( { – 2,\,5} \right)\) and \(G\left( { – 3,\, – 4} \right)\) when reflected in the \(y\)-axis.
Ans: When a point is reflected in the \(y\)-axis, the coordinate remains the same while the \(x\) coordinate changes its sign.
\(D\left( {4,5} \right) \to D’\left( { – 4,\,5} \right)\) under reflection in the \(y\)-axis.
\(E\left( {7,\, – 3} \right) \to E’\left( { – 7,\, – 3} \right)\) under reflection in the \(y\)-axis.
\(F\left( { – 2,\,5} \right) \to F’\left( {2,\,5} \right)\) under reflection in the \(y\)-axis.
\(G\left( { – 3,\, – 4} \right) \to D’\left( {3,\, – 4} \right)\) under reflection in the \(y\)-axis.

Q.3. Reflect \(\Delta ABC\) in the \(x\)-axis given \(A\left( { – 4,\,6} \right),\,B\left( { – 8,\,2} \right)\) and \(C\left( { – 2,\,3} \right).\) What do you observe?
Ans: Mark the points \(A\left( { – 4,\,6} \right),\,B\left( { – 8,\,2} \right)\) and \(C\left( { – 2,\,3} \right)\) on a graph paper. Join \(AB,\,BC\) and \(CA\) to get \(\Delta ABC.\) Mark the points \(A’\left( { – 4,\, – 6} \right),\,B’\left( { – 8,\, – 2} \right)\) and \(C’\left( { – 2,\, – 3} \right).\) Join \(A’B’,\,B’C’\) and \(C’A’.\)

\(\Delta A’B’C’\) is the reflection of \(\Delta ABC\) in the \(x\)-axis. We observe that \(ABC\) is congruent to \(\Delta A’B’C’.\) We also see that \(\Delta ABC\) and \(\Delta A’B’C’\) are symmetrical about the \(x\)-axis.

Q.4. \(O\) is \(\left( {0,\,0} \right)\) and \(A\) is \(\left( {3,\,1} \right).\) If \(OA\) is rotated through \({90^{\rm{o}}}\) about \(O\) in the positive (anti-clockwise) direction to \(OA’,\) find the coordinates of \(A’.\)
Ans: In the figure, when \(OA\) is rotated through \({90^{\rm{o}}}\) in the positive direction, we arrive at \(OA’.\,\angle AOA’ = {90^{\rm{o}}}.\)

We can observe that \(A’\) is \(\left( { – 1,\,3} \right),\) since \(OA = OA’.\) So, the coordinate of \(A’\) are \(\left( { – 1,\,3} \right).\)

Q.5. Write down the coordinates of the images of the point \(A\left( {2,\,3} \right)\) when reflected on the origin.
Ans: \(A\left( {2,\,3} \right) \to A’\left( { – 2,\, – 3} \right)\) when reflected in the origin \(\left( {0,\,0} \right).\)

Summary

In this article, we learned about reflection definition, the reflection of a point in a line, reflections in the coordinate plane, reflection over \(X\)-axis, reflection over \(Y\)-axis, reflection over \(Y = X,\) reflection in a point, reflection in origin \(\left( {0,\,0} \right).\) solved examples on reflection of point using graph paper and FAQs on reflection of point using graph paper. The learning outcome of this article is we understood how to reflect the point in the coordinate plane. Reflections symmetry is a concept utilised in the design of astronomical telescopes.

Learn about Reflection Symmetry here

FAQs on Reflection of Point using Graph Paper

Q.1. How do you find the reflection of a point on a graph?
Ans: The \(x\)-coordinate remains the same when a point is reflected across the \(x\)-axis, while the \(y\)-coordinate is turned into the opposite (its sign is changed).
The \(y\)-coordinate remains the same when a point is reflected across the \(y\)-axis, while the \(x\)-coordinate is changed into the opposite (its sign is changed).

Q.2. What is the reflection of a point on a graph?
Ans: When a figure is built around a single point known as the point of reflection or its centre, it becomes a reflection point. On the other side of every point in the diagram, another point is discovered precisely opposite it. The size and shape of the figure do not change when it is reflected.

Q.3. How do you write a reflection over the \(y\)-axis?
Ans: The \(y\)-coordinate remains the same when a point is reflected across the \(y\)-axis, while the \(x\)-coordinate is changed into the opposite (its sign is changed).

Q.4. How do you write a reflection over the \(x\)-axis?
Ans: The \(x\)-coordinate remains the same when a point is reflected across the \(x\)-axis, while the \(y\)-coordinate is turned into the opposite (its sign is changed).
When graphing, if you forget the rules for reflections, fold the paper along the \(x\)-axis (the line of reflection) to see where the new figure will be.
The point \(\left( {x,\,y} \right)\) reflected across the \(x\)-axis is \(\left( {x,\, – y} \right).\)

Q.5. What is a line of reflection?
Ans: A perpendicular bisector is a reflecting line. The reflecting line is the perpendicular bisector of all segments that connect pre-image locations to their corresponding image points when a figure is reflected.