Notes
Coordinate rules
| Transformation | Rule |
|---|---|
| Translation | $(x, y) \to (x + a, y + b)$ |
| Reflect over $x$-axis | $(x, y) \to (x, -y)$ |
| Reflect over $y$-axis | $(x, y) \to (-x, y)$ |
| Reflect over $y = x$ | $(x, y) \to (y, x)$ |
| Rotate $90^\circ$ counterclockwise | $(x, y) \to (-y, x)$ |
| Rotate $180^\circ$ | $(x, y) \to (-x, -y)$ |
| Rotate $270^\circ$ counterclockwise | $(x, y) \to (y, -x)$ |
| Dilate by $k$ from origin | $(x, y) \to (kx, ky)$ |
Rigid motions and dilations
- Translations, reflections and rotations are rigid motions: they keep lengths and angles, so the image is congruent.
- Reflections reverse orientation. Translations and rotations keep it.
- A dilation multiplies lengths by $k$ and keeps angles, so the image is similar.
Compositions and symmetry
- Do a composition in order. Two reflections over parallel lines make a translation; over intersecting lines, a rotation.
- A regular $n$-gon has $n$ lines of symmetry and its smallest rotation symmetry is $\displaystyle \frac{360^\circ}{n}$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$(x, y) \to (-y, x)$ gives $(-2, 1)$.
A reflection flips the figure, reversing orientation.
Translate: $(-1, 1)$. Reflect over the $x$-axis: $(-1, -1)$.
New sides are 10 and 15, so the perimeter is $2(10 + 15) = 50$.
$360^\circ \div 8 = 45^\circ$.
Two reflections over intersecting lines give a rotation of twice the angle between them.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A video game designer places a spaceship shaped like triangle $ABC$ with $A(1, 1)$, $B(4, 1)$ and $C(1, 5)$ on a coordinate grid.
(a) The ship moves 3 units left and 2 units up. Write the rule and give the new coordinates.
Show the answer and scoring guide
Rule: $(x, y) \to (x - 3, y + 2)$. $A'(-2, 3)$, $B'(1, 3)$, $C'(-2, 7)$.
Check off the points you earned:
(b) Instead, the original ship is rotated $180^\circ$ about the origin. Give the new coordinates and explain whether the ship changes size.
Show the answer and scoring guide
$A'(-1, -1)$, $B'(-4, -1)$, $C'(-1, -5)$. A rotation is a rigid motion, so the size and shape stay the same.
Check off the points you earned:
(c) For a power-up, the original ship is dilated by a scale factor of 2 centered at the origin. Find the length of side $\overline{BC}$ before and after.
Show the answer and scoring guide
Before: $\displaystyle BC = \sqrt{(1 - 4)^2 + (5 - 1)^2} = \sqrt{9 + 16} = 5$. After: $B'(8, 2)$ and $C'(2, 10)$, so $B'C' = 2 \cdot 5 = 10$.
Check off the points you earned: