Why a Temple's Reflection in a Pond Flips Upside Down but Not Left to Right

When a point is reflected in a mirror line, its image lands on the other side of the line, at the same distance from it. The line joining the point and its image is perpendicular to the mirror line, and the mirror line bisects it.

On a coordinate grid, this means reflection simply changes the sign of one or both coordinates, or shifts them about a fixed line. This chapter covers reflections in the axes, the origin and the lines $x=a$ and $y=a$, invariant points, and successive reflections.

Reflection in the x-axis changes the sign of the y-coordinate, reflection in the y-axis changes the sign of the x-coordinate, and reflection in the origin changes both signs.

- In the x-axis ($M_x$): $(x,y)\to (x,-y)$
- In the y-axis ($M_y$): $(x,y)\to (-x,y)$
- In the origin ($M_O$): $(x,y)\to (-x,-y)$

Worked example. Reflect $P(3,-5)$.

$$M_x:(3,5)M_y:(-3,-5)M_O:(-3,5)$$

Worked example. Reflect $Q(-4,0)$ in the y-axis: $(4,0)$. In the x-axis: $(-4,0)$, which is unchanged, because $Q$ lies on the x-axis.

An everyday example. A temple beside a still pond appears upside down in the water, but its left and right sides stay the same — just as reflection in the x-axis changes only the vertical coordinate.

The substance. Reflecting in the x-axis and then the y-axis gives the same result as reflecting in the origin, in either order.

**In the line $x=a$, the y-coordinate stays and the x-coordinate becomes $2a-x$; in the line $y=a$, the x-coordinate stays and the y-coordinate becomes $2a-y$.

- In $x=a$**: $(x,y)\to (2a-x,y)$
- **In $y=a$**: $(x,y)\to (x,2a-y)$

Why. The mirror line must be exactly halfway between the point and its image, so $a=\frac{x+x^{\prime }}{2}$, giving $x^{\prime }=2a-x$.

Worked examples.

$$(5,2)\text{ in }x=3:(2\times 3-5,2)=(1,2)$$
$$(-1,4)\text{ in }x=2:(4+1,4)=(5,4)$$
$$(4,-3)\text{ in }y=1:(4,2+3)=(4,5)$$
$$(0,6)\text{ in }y=-2:(0,-4-6)=(0,-10)$$

An everyday example. Folding a sheet of graph paper along a ruled line and pressing a wet ink dot through shows exactly where its image lands.

The link. **The x-axis is the line $y=0$**, so the rule gives $(x,0-y)=(x,-y)$, matching the earlier result.

Invariant points are points that stay in the same place after a reflection, which are exactly the points lying on the mirror line; the mirror line itself is the perpendicular bisector of the segment joining a point and its image.

Invariant points.

- **Under $M_x$**: all points $(x,0)$ on the x-axis, such as $(3,0)$
- **Under $M_y$**: all points $(0,y)$, such as $(0,-7)$
- **Under reflection in $x=2$**: all points with $x=2$, such as $(2,-7)$
- **Under $M_O$: only the origin** $(0,0)$

Naming the mirror line.

- $(5,3)\to (5,-3)$: the x-axis
- $(-2,4)\to (2,4)$: the y-axis
- $(3,-1)\to (-3,1)$: the origin
- $(1,6)\to (7,6)$: the line $x=\frac{1+7}{2}=4$
- $(2,-5)\to (2,9)$: the line $y=\frac{-5+9}{2}=2$

An everyday example. When a paper rangoli stencil is folded and cut, the points on the fold do not move when the paper is opened — they are the invariant points.

The substance. Reflection in the origin is not a reflection in a line, which is why it has only one invariant point.

Reflect the point step by step, writing each image's coordinates, plot all the points, join them in order, and use lengths of sides to name the figure.

Worked example 1. $A(4,3)$ is reflected in the x-axis to $B$; $B$ is reflected in the y-axis to $C$; $A$ is reflected in the y-axis to $D$.

$$B=(4,-3)C=(-4,-3)D=(-4,3)$$

**$AB$ is vertical with length $6$ and $BC$ is horizontal with length $8$**, so $ABCD$ is a rectangle:

$$\text{area}=8\times 6=48\text{ square units}\text{perimeter}=2(8+6)=28\text{ units}$$

Worked example 2. $P(2,5)$ is reflected in the y-axis to $Q$, and $P$ is reflected in the origin to $R$.

$$Q=(-2,5)R=(-2,-5)$$

**$PQ=4$ horizontally and $QR=10$ vertically**, meeting at a right angle, so $PQR$ is a right-angled triangle with area $\frac{1}{2}\times 4\times 10=20$ square units.

An everyday example. A kolam drawn at a doorway often repeats one shape by reflecting it across two perpendicular lines, filling all four quarters.

The substance. Two reflections in perpendicular lines through the origin give a half-turn about the origin.