### Reflection

If we stand in front of a mirror, our images come closer to the mirror and vice versa, it is known as reflection. In reflection, the mirror is represented by a line, which is called the axis of reflection. The properties of reflection are:

1. Lateral inversion of each other
2. Equidistant from the mirror line
3. Congruent to each other

#### Reflection using Co-ordinates

By using the coordinates, we can reflect a given object about the x-axis, y-axis, line y = x, line y= x etc.

• Reflection in X-axis: When we reflect a point across the x-axis, the x co-ordinate remains the same, but y co-ordinate is transformed into negative.
The reflection of the point (x, y) across the x-axis is the point (x,−y).
P(x, y) → P'(x, -y)
or, x-axis (x, y) = (x,−y)

• Reflection in Y-axis: When we reflect a point across the y-axis, then y co-ordinator remains the same, but the x co-ordinate is transformed into negative.
The reflection of the point (x, y) across the y-axis is the point (−x, y).
P(x, y) → P'(−x, y)
or, y-axis (x, y) = (−x, y)

• Reflecting in the line y = x: When we reflect a point across the line y = x, the x co-ordinate and then y co-ordinate change place.
The reflection of the point (x, y) across the line y = x is the point (y, x).
P(x, y) → P'(y, x)
or, y = x (x, y) = (y, x)

• Reflection in line y = −x: When we reflect a point across the line y = −x, the x co-ordinate and then y co-ordinate change places and are negated or the signs are changes.
The reflection point (x, y) across the line y = −x is the point (−y,−x)
P(x, y) → P'(−y,−x)

• The reflection of the point (x, y) across the x-axis is the point (x, −y) = P(x, y) → P'(x, y)
• The reflection of the point (x, y) across the y-axis is the point (−x, y) = P(x, y) → P'(−x, y)
• The reflection of the point (x, y) across the line y = x is the point (y, x) = P(x, y) → P'(y, x)
• The reflection of the point (x, y) across the line y = −x is the point (−y, −x) = P(x, y) → P'(−y, −x)

Solution;

Under reflection about x-axis,

P(x, y) $$\rightarrow$$ P'(x, -y)

$$\therefore$$ A(1, 3) $$\rightarrow$$ A'(1, -3)

B(4, 5) $$\rightarrow$$ B'(4, -5) and

C(6, 2) $$\rightarrow$$ C'(6, -2)

$$\triangle$$ABC and its image $$\triangle$$A'B'C' are shown on the graph.

Solution;

Under reflection about y-axis,

P(x, y) $$\rightarrow$$ P'(-x, y)

$$\therefore$$ A(2, 3) $$\rightarrow$$ A'(-2, 3)

B(4, 5) $$\rightarrow$$ B'(-4, 5) and

C(6, 2) $$\rightarrow$$ C'(-6, 2)

$$\triangle$$ABC and its image $$\triangle$$A'B'C' are shown on the graph.

Solution;

Under reflection about x-axis,

P(x, y) $$\rightarrow$$ P'(x, -y)

$$\therefore$$ P(3, 1) $$\rightarrow$$ P'(3, -1)

Q(5, 4) $$\rightarrow$$ Q'(5, -4) and

R(2, 6) $$\rightarrow$$ R'(2, -6)

$$\triangle$$PQR and its image $$\triangle$$P'Q'R' are shown on the graph.

Solution:

Under reflection about x-axis

P(x, y) $$\rightarrow$$ P'(x, -y)

$$\therefore$$ A(-4, 5) $$\rightarrow$$ A'(-4, -5) and

B(6, -2) $$\rightarrow$$ B'(6, 2)

Line AB and its image A'B' are shown on the graph.

Solution;

Under reflection about y-axis,

P(x, y) $$\rightarrow$$ P'(-x, y)

$$\therefore$$ X(1, 3) $$\rightarrow$$ X'(-1, 3) and

Y(4, 5) $$\rightarrow$$ Y'(-4, 5)

Line XY and its image X'Y' are shown on the graph.

Solution:

Under reflection about x-axis

P(x, y) $$\rightarrow$$ P'(x, -y)

$$\therefore$$ A(-5, -2) $$\rightarrow$$ A'(-5, 2) and

B(2, 3) $$\rightarrow$$ B'(2, -3)

Line AB and its image A'B' are shown on the graph.

Solution:

Under reflection about y-axis

P(x, y) $$\rightarrow$$ P'(-x, y)

$$\therefore$$ A(-5, -2) $$\rightarrow$$ A'(5, -2) and

B(2, 3) $$\rightarrow$$ B'(-2, 3)

Line AB and its image A'B' are shown on the graph.

Solution:

Let AB be the given point, then

A(3, 4) and B (8, 10)

Under reflection about y-axis

P(x, y) $$\rightarrow$$ P'(-x, y)

$$\therefore$$ A(3, 4) $$\rightarrow$$ A'(-3, 4) and

B(8, 10) $$\rightarrow$$ B'(-8, 10)

line AB and its image A'B' are shown on the graph.

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#### Discussions about this note

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##### Bibek

what is reflection?

##### shivam

what is reflection

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