Reflection of geometric shapes

Learn what a point is, see how it’s defined, then watch the equation of the circle get derived — one step at a time.

Introduction to Reflection

A reflection is a transformation that flips a geometric figure across a line or point, called the line or point of reflection, producing a mirror image. Different types of reflections are useful depending on the axis or line of reflection being used.

In this topic, we will learn the following standard types of reflection in the Cartesian plane:

Reflection in the X-axis –> transforms a point $(x, y)$ to $(x, -y)$ by negating its ordinate.
Reflection in the Y-axis –> transforms a point $(x, y)$ to $(-x, y)$ by negating its abscissa.
Reflection in the line $x = a$ –> transforms a point $(x, y)$ to $(2a - x, y)$ by reflecting across the vertical line $x = a$.
Reflection in the line $y = b$ –> transforms a point $(x, y)$ to $(x, 2b - y)$ by reflecting across the horizontal line $y = b$.
Reflection in the line $y = x$ –> swaps the coordinates, transforming a point $(x, y)$ to $(y, x)$.
Reflection in the line $y = -x$ –> transforms a point $(x, y)$ to $(-y, -x)$ by swapping and negating both coordinates.

Properties of Reflection

The object and its image are at equal distances from the axis of reflection.
The line joining the object and its image is perpendicular to the axis of reflection.
The object and its reflected image are congruent.
The object and its image are inverted.
Points lying on the given axis of reflection remain invariant (unchanged).

Reflection under X-Axis

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Reflection under X-Axis

Definition:

Reflection in the X-axis flips a figure across the X-axis like a mirror. Each point keeps its distance from the X-axis but moves to the opposite side. The x-coordinate stays the same and the y-coordinate changes sign.

1. Procedure:

Take graph and draw the given triangle ΔABC. Reflect it across the X-axis on the same drawing. Write the coordinates of the reflected triangle ΔA'B'C' obtained by this reflection and present the drawing. Then compare the coordinates of ΔABC and its image ΔA'B'C' and present your conclusion.

2. Original Points:

Triangle ABC: A(2,6), B(6,4), C(3,2)

3. Points (A and A', B and B', C and C'):

A(2,6)Reflection about X-Axis: (x, -y)A'(2,-6)

B(6,4)Reflection about X-Axis: (x, -y)B'(6,-4)

C(3,2)Reflection about X-Axis: (x, -y)C'(3,-2)

4. Formula:

P(x, y) — Re: X-axis → P'(x, -y)

5. Conclusion:

The x-coordinate of every point stays the same, while the y-coordinate changes its sign. So ΔABC moves from above the X-axis to below it, giving ΔA'B'C' with A'(2,-6), B'(6,-4), C'(3,-2). Its size and shape stay the same.

Reflection under Y-Axis

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Reflection under Y-Axis

Definition:

Reflection in the Y-axis flips a figure across the Y-axis like a mirror. Each point keeps its distance from the Y-axis but moves to the opposite side. The y-coordinate stays the same and the x-coordinate changes sign.

1. Procedure:

Take graph and draw the given triangle ΔABC. Reflect it across the Y-axis on the same drawing. Write the coordinates of the reflected triangle ΔA'B'C' obtained by this reflection and present the drawing. Then compare the coordinates of ΔABC and its image ΔA'B'C' and present your conclusion.

2. Original Points:

Triangle ABC: A(4,6), B(6,4), C(1,2)

3. Points (A and A', B and B', C and C'):

A(4,6)Reflection about Y-Axis: (-x, y)A'(-4,6)

B(6,4)Reflection about Y-Axis: (-x, y)B'(-6,4)

C(1,2)Reflection about Y-Axis: (-x, y)C'(-1,2)

4. Formula:

P(x, y) — Re: Y-axis → P'(-x, y)

5. Conclusion:

The y-coordinate of every point stays the same, while the x-coordinate changes its sign. So ΔABC moves from the right side to the left side across the Y-axis, giving ΔA'B'C' with A'(-4,6), B'(-6,4), C'(-1,2). Its size and shape stay the same.

Reflection on the line y = x

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Reflection on the line y = x

Definition:

Reflection in the line y = x flips a figure across that diagonal line like a mirror. Each point keeps its distance from the line but moves to the opposite side. The x-coordinate and the y-coordinate swap their places.

1. Problem:

In the given graph, when the points, A(5, 3) and B(8, 4) of line AB are reflected across the line y = x, the reflected points A' and B' of line A'B' are shown in the graph. What are the coordinates of A' and B'? Compare and draw a conclusion.

2. Procedure:

Reflect the points A(5, 3) and B(8, 4) across the line y = x, write the coordinates of the reflected points A' and B' of line AB and discuss on your conclusion.

3. Points (A and A', B and B'):

A(5,3)y = xA'(3,5)

B(8,4)y = xB'(4,8)

4. Formula:

Thus, P(x, y) — Re: y = x → P'(y, x)

5. Conclusion:

The x-coordinate and y-coordinate of every point are interchanged. So AB moves to the opposite side of the line y = x, giving A'B' with A'(3,5) and B'(4,8). Its length and shape stay the same.

Reflection on the line y = – x

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Reflection on the line y = – x

Definition:

Reflection in the line y = −x flips a figure across that diagonal line like a mirror. Each point keeps its distance from the line but moves to the opposite side. The x-coordinate and the y-coordinate swap their places and both change their signs.

1. Problem:

In the given graph, when the points, A(5, 2) and B(8, 4) of line AB are reflected across the line y = −x, the reflected points A' and B' of line A'B' are shown in the graph. What are the coordinates of A' and B'? Compare and draw a conclusion.

2. Procedure:

Reflect the points A(5, 2) and B(8, 4) across the line y = −x, write the coordinates of the reflected points A' and B' of line AB and discuss on your conclusion.

3. Points (A and A', B and B'):

A(5,2)y = −xA'(−2, −5)

B(8,4)y = −xB'(−4, −8)

4. Formula:

Thus, P(x, y) — Re: y = −x → P'(−y, −x)

5. Conclusion:

The x-coordinate and y-coordinate of every point are interchanged and both change their sign. So AB moves to the opposite side of the line y = −x, giving A'B' with A'(−2,−5) and B'(−4,−8). Its length and shape stay the same.

Reflection on the line x = a

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Reflection on the line x = a

Definition:

Reflection in the line x = a flips a figure across the vertical line x = a like a mirror. Each point keeps its distance from the line but moves to the opposite side. The y-coordinate stays the same and the x-coordinate becomes 2a − x.

1. Problem:

In the given graph, when the points, E(1, 4) and F(−1, 3) of line EF are reflected across the line x = 3, the reflected points E' and F' of line E'F' are shown in the graph. What are the coordinates of E' and F'? Compare and draw a conclusion.

2. Procedure:

Reflect the points E(1, 4) and F(−1, 3) across the line x = 3, write the coordinates of the reflected points E' and F' of line EF and discuss on your conclusion.

3. Points (E and E', F and F'):

E(1,4)x = 3E'(5,4) = E'(2 × 3 − 1, 4)

F(−1,3)x = 3F'(7,3) = F'(2 × 3 − (−1), 3)

4. Formula:

Thus, P(x, y) — Re: x = a → P'(2a − x, y)

5. Conclusion:

The y-coordinate of every point stays the same, while the x-coordinate becomes 2a − x. So EF moves to the opposite side of the line x = 3, giving E'F' with E'(5,4) and F'(7,3). Its length and shape stay the same.

Reflection on the line y = b

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Reflection on the line y = b

Definition:

Reflection in the line y = b flips a figure across the horizontal line y = b like a mirror. Each point keeps its distance from the line but moves to the opposite side. The x-coordinate stays the same and the y-coordinate becomes 2b − y.

1. Problem:

In the given graph, when the points, M(4, 5) and N(8, 4) of line MN are reflected across the line y = 3, the reflected points M' and N' of line M'N' are shown in the graph. What are the coordinates of M' and N'? Compare and draw a conclusion.

2. Procedure:

Reflect the points M(4, 5) and N(8, 4) across the line y = 3, write the coordinates of the reflected points M' and N' of line MN and discuss on your conclusion.

3. Points (M and M', N and N'):

M(4,5)y = 3M'(4,1) = M'(4, 2 × 3 − 5)

N(8,4)y = 3N'(8,2) = N'(8, 2 × 3 − 4)

4. Formula:

Thus, P(x, y) — Re: y = b → P'(x, 2b − y)

5. Conclusion:

The x-coordinate of every point stays the same, while the y-coordinate becomes 2b − y. So MN moves to the opposite side of the line y = 3, giving M'N' with M'(4,1) and N'(8,2). Its length and shape stay the same.

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