Reflection — chapter notes

• A reflection in a mirror line $l$ sends each point $P$ to its image $P'$ on the other side of $l$, at the same distance from it.
• The mirror line is the perpendicular bisector of $PP'$: $PP'$ meets $l$ at right angles, and $l$ passes through the midpoint of $PP'$.
• A point and its image are $2k$ units apart when the point is $k$ units from the mirror.
• A point on the mirror is its own image (it is invariant).

Example: $P(4, 3)$ in the line $y = 1$: $P$ is $2$ units above the mirror, so $P'$ is $(4, -1)$, $2$ units below it.

• In the $x$-axis ($y = 0$, written $M_x$): keep $x$, change the sign of $y$.
• In the $y$-axis ($x = 0$, written $M_y$): change the sign of $x$, keep $y$.

Example: $A(-5, 2)$ goes to $(-5, -2)$ in the $x$-axis and to $(5, 2)$ in the $y$-axis; $B(0, -3)$ lies on the $y$-axis, so it is invariant under $M_y$.

• In the origin (written $M_O$): change the signs of both coordinates. The origin is the midpoint of $PP'$, and $P$, $O$, $P'$ lie on one straight line.

• Reflection in the $x$-axis followed by reflection in the $y$-axis (in either order) has the same effect as $M_O$.
• Only the origin is invariant under $M_O$.

Example: the image of $(3, 2)$ in the origin is $(-3, -2)$; and if the image of $P$ in the $y$-axis is $(-6, -1)$, then $P$ is $(6, -1)$ and its image in the origin is $(-6, 1)$.

• In $x = a$ (a vertical mirror): keep $y$; a point $x - a$ to the right goes $x - a$ to the left, so the new $x$-coordinate is $a - (x - a) = 2a - x$.
• In $y = b$ (a horizontal mirror): keep $x$; the new $y$-coordinate is $2b - y$.

• With $a = 0$ or $b = 0$ these become the rules for the axes.

Example: $R(7, -2)$ goes to $(2 \times 3 - 7, -2) = (-1, -2)$ in $x = 3$ and to $(7, 4)$ in $y = 1$.

• The point from its image: a reflection undoes itself, so reflect the image in the same mirror.
• The mirror from a point and its image: if the $x$-coordinates are equal, the mirror is horizontal, $y = \dfrac{y_1 + y_2}{2}$; if the $y$-coordinates are equal, it is vertical, $x = \dfrac{x_1 + x_2}{2}$; if both coordinates change sign, it is the origin.
• Unknown coordinates: write the image by the rule, then equate $x$-coordinates and $y$-coordinates separately.

Example: the image of $(2a - 3, 5)$ in the $y$-axis is $(-7, b + 2)$: $3 - 2a = -7$, so $a = 5$; $b + 2 = 5$, so $b = 3$.

• A point is invariant under a reflection if its image is the point itself.
• For a mirror line, the invariant points are exactly the points on that line: $(x, 0)$ for the $x$-axis, $(0, y)$ for the $y$-axis, $(a, y)$ for $x = a$, $(x, b)$ for $y = b$.
• For the origin, only $O(0, 0)$ is invariant.
• A point invariant under two reflections in perpendicular lines is their point of intersection.

Example: $D(-2, -2)$ is invariant under reflection in both $x = -2$ and $y = -2$.

• Two reflections: $x$-axis then $y$-axis (either order) $=$ reflection in the origin; the same mirror twice $=$ no change.
• A figure: reflect each vertex and join the images in the same order. The image is congruent (same lengths, angles, area) but the order of the vertices is reversed.
• Naming the figure: horizontal and vertical sides give parallel sides and right angles at a glance; then compare side lengths (rectangle, square, isosceles trapezium, kite, right-angled or isosceles triangle).
• Area: use horizontal or vertical sides as bases and heights; trapezium $\dfrac{1}{2}(a + b)h$; kite $\dfrac{1}{2} d_1 d_2$.

Example: $A(5, 2)$ with its images in the $y$-axis, the origin and the $x$-axis makes a rectangle $10$ by $4$: area $40$ square units, perimeter $28$ units.