Notes
Rigid motions
- Translation: $(x, y) \to (x + a, y + b)$. Reflection: over the $x$-axis $(x, -y)$, over the $y$-axis $(-x, y)$, over $y = x$ $(y, x)$.
- Rotation about the origin (counterclockwise): $90^\circ$ $(-y, x)$; $180^\circ$ $(-x, -y)$; $270^\circ$ $(y, -x)$.
- Rigid motions keep lengths and angles; dilations do not.
Congruence
- Figures are congruent if rigid motions map one onto the other. Corresponding parts are equal.
- Triangle criteria: SSS, SAS, ASA, AAS, HL. AAA and SSA do not prove congruence.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Changing only the sign of $x$ flips the point across the $y$-axis.
$-2 + 6 = 4$ and $-1 + 3 = 2$.
A $90^\circ$ clockwise turn is $(x, y) \to (y, -x)$, so $(6, -2) \to (-2, -6)$.
Letters match in order: $J \leftrightarrow M$, $K \leftrightarrow N$, $L \leftrightarrow P$.
Two angles and a non-included side is AAS.
The point is 3 units above the line, so the image is 3 units below: $y = 2 - 3 = -1$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Triangle $ABC$ has vertices $A(1, 1)$, $B(4, 1)$ and $C(1, 5)$.
(a) Translate the triangle by $(x, y) \to (x - 5, y + 2)$. Give the image vertices.
Show the answer and scoring guide
$A'(-4, 3)$, $B'(-1, 3)$, $C'(-4, 7)$.
Check off the points you earned:
(b) Reflect the original triangle over the $x$-axis. Give the image vertices.
Show the answer and scoring guide
$A''(1, -1)$, $B''(4, -1)$, $C''(1, -5)$.
Check off the points you earned:
(c) Explain why both images are congruent to $ABC$, and find the length of $A''C''$.
Show the answer and scoring guide
Translations and reflections are rigid motions, so they preserve side lengths and angle measures. $A''C''$ is vertical from $(1, -1)$ to $(1, -5)$, so its length is $4$, the same as $AC$.
Check off the points you earned: