1. Reflection
A reflection flips a figure over a line (the axis of reflection) to create a mirror image. The object and its image are equidistant from the reflection line.
| Axis of Reflection |
Object Point $P(x, y)$ |
Image Point $P'(x', y')$ |
| X-axis ($y = 0$) |
$(x, y)$ |
$(x, -y)$ |
| Y-axis ($x = 0$) |
$(x, y)$ |
$(-x, y)$ |
| Line $y = x$ |
$(x, y)$ |
$(y, x)$ |
| Line $y = -x$ |
$(x, y)$ |
$(-y, -x)$ |
| Line $x = h$ |
$(x, y)$ |
$(2h - x, y)$ |
| Line $y = k$ |
$(x, y)$ |
$(x, 2k - y)$ |
Example: Reflect the point $A(3, 4)$ over the line $y = x$.
Solution: Using the rule $P(x, y) \to P'(y, x)$, the image is $A'(4, 3)$.
2. Rotation
Rotation turns a figure about a fixed point (center of rotation) through a specific angle. A positive angle indicates counter-clockwise direction, while a negative angle indicates clockwise direction.
Rotation about the Origin $(0, 0)$
| Angle of Rotation |
Object Point $P(x, y)$ |
Image Point $P'(x', y')$ |
| $+90^\circ$ (Quarter turn counter-clockwise) |
$(x, y)$ |
$(-y, x)$ |
| $-90^\circ$ (Quarter turn clockwise) |
$(x, y)$ |
$(y, -x)$ |
| $\pm180^\circ$ (Half turn) |
$(x, y)$ |
$(-x, -y)$ |
Rotation about a Point $(a, b)$
When the center of rotation is not the origin, we logically shift the origin to $(a, b)$, apply the standard rotation, and then shift back. This gives us the following direct formulas:
| Angle of Rotation |
Object Point $P(x, y)$ |
Image Point $P'(x', y')$ about $(a, b)$ |
| $+90^\circ$ |
$(x, y)$ |
$(-y + b + a, x - a + b)$ |
| $-90^\circ$ |
$(x, y)$ |
$(y - b + a, -x + a + b)$ |
| $\pm180^\circ$ |
$(x, y)$ |
$(2a - x, 2b - y)$ |
Example (Not on Origin): Rotate point $B(5, 2)$ through $+90^\circ$ about the center $(1, 3)$.
Solution: Here, $x=5, y=2$ and $a=1, b=3$.
Using the rule $P'(-y + b + a, x - a + b)$:
$x' = -2 + 3 + 1 = 2$
$y' = 5 - 1 + 3 = 7$
The image is $B'(2, 7)$.
3. Translation
Translation slides a figure in a specific direction by a specific distance without rotating or flipping it. It is defined by a translation vector $T = \begin{pmatrix} a \\ b \end{pmatrix}$.
If an object point is $P(x, y)$ and the translation vector is $T = \begin{pmatrix} a \\ b \end{pmatrix}$, the image point is:
$$P'(x + a, y + b)$$
Example: Translate point $C(1, -3)$ by the vector $T = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$.
Solution: $x' = 1 + 4 = 5$ and $y' = -3 + 2 = -1$. The image is $C'(5, -1)$.
4. Enlargement (Dilation)
Enlargement changes the size of a figure without altering its shape. It requires a center of enlargement and a scale factor ($k$).
| Center of Enlargement |
Scale Factor |
Image Point Rule |
| Origin $(0, 0)$ |
$k$ |
$P'(kx, ky)$ |
| Point $(a, b)$ |
$k$ |
$P'(k(x - a) + a, k(y - b) + b)$ |
Example (Center at $(a, b)$): Enlarge point $D(4, 5)$ with center $(2, 1)$ and scale factor $k = 2$.
Solution: Using $P'(k(x - a) + a, k(y - b) + b)$:
$x' = 2(4 - 2) + 2 = 2(2) + 2 = 6$
$y' = 2(5 - 1) + 1 = 2(4) + 1 = 9$
The image is $D'(6, 9)$.