Geometry Workbook — Chapter 3: Symmetry and Transformations

SOL G.RLT.3 (a, b, c, d) · Companion to Textbook Chapter 3

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 110.

PAGE 1 — Chapter opener

Chapter 3 · Symmetry and Transformations

Standard G.RLT.3 (a, b, c, d)

In this chapter you will:

Locate, count, and draw every line of symmetry of a figure
Decide whether a figure has line symmetry, point symmetry, both, or neither
Apply the coordinate rules for translations, the four named reflections, rotations of $90°$ , $180°$ , $270°$ about the origin, and dilations from the origin
Find a preimage by running a rule backwards
Name the transformation from a preimage-and-image pair, including a combination of two

Words to know: line of symmetry · point symmetry · preimage · image · translation · reflection · rotation · dilation · scale factor · rigid motion · orientation · composition

Convention: rotations are counterclockwise about the origin unless a problem says otherwise. Image vertices carry a prime: $A \rightarrow A'$ ; a second transformation adds a second prime.

PAGE 2 — Lines of symmetry

3.1 Lines of Symmetry

FIGURE: fig1-lines-of-symmetry-in-polygons.png (full width)

Fill in the blanks.

A line of symmetry is a line you could ____________________ the figure along so the two halves land ____________________ on each other.

Square: ______ lines · Rectangle: ______ lines
Isosceles trapezoid: ______ line. Where is it? ____________________
Parallelogram: ______ lines. Why does a diagonal not work?

PAGE 3 — Counting on regular polygons

The n-gon Rule

FIGURE: fig2-counting-lines-of-symmetry.png (full width)

Fill in the blank. A regular polygon with $n$ sides has exactly ______ lines of symmetry.

Regular pentagon: ______ Regular hexagon: ______
Describe where the hexagon's six lines are.
State the rule for a regular $n$ -gon. ____________________

Complete the reference table.

Figure	Lines of symmetry
Square	______
Rectangle (not square)	______
Rhombus (not square)	______
Parallelogram (neither)	______
Isosceles trapezoid	______
Kite	______
Equilateral triangle	______
Scalene triangle	______

PAGE 4 — Symmetry in context

Real Designs

FIGURE: fig3-symmetry-in-context.png (full width)

Stop sign: ______ lines Window arch: ______ lines
Application. A tile is a regular hexagon with a pattern repeating identically in all six sections. Lines of symmetry: ______
Application. A road sign is an equilateral triangle with one word printed horizontally across it. Lines of symmetry once the word is added: ______ Explain:

PAGE 5 — Practice · counting and drawing

Practice

a) equilateral triangle ______ b) isosceles (not equilateral) ______ c) scalene ______ d) regular octagon ______
a) square ______ b) rhombus (not square) ______ c) rectangle (not square) ______ d) kite ______
A regular polygon has $12$ lines of symmetry. Sides: ______
A regular $15$ -gon: ______ lines · how many run vertex to vertex? ______
Sketch a rhombus, draw all lines of symmetry, and say what they are.
Sketch a rectangle, draw all lines of symmetry, and say what they are.
Draw a figure with exactly one line of symmetry, and a different figure with exactly two.

PAGE 6 — Reasoning and exit · Lesson 3.1

Think It Through

Error analysis. A student says a parallelogram has two lines of symmetry because its diagonals cut it into congruent triangles. What is the error?
Reasoning. Why does a regular polygon with an odd number of sides have no line joining two vertices?
Reasoning. Describe all of a circle's lines of symmetry in one sentence.
Regular nonagon: ______ lines
Rectangle (not square): ______ lines · where? ____________________
Does a parallelogram (neither rectangle nor rhombus) have lines of symmetry? Explain in one sentence.

PAGE 7 — Two kinds of symmetry

3.2 Line and Point Symmetry

FIGURE: fig4-line-versus-point-symmetry.png (full width)

Fill in the blanks.

Line symmetry — some ____________________ folds the figure onto itself.

Point symmetry — some ____________________ is a center about which a ______ turn carries the figure onto itself.

Which figure has both kinds? ____________________
Which has point symmetry but no lines? ____________________
Which has one line and no point symmetry? ____________________
State the difference in one sentence each.

Line: ____________________ Point: ____________________

PAGE 8 — Point symmetry is a half turn

The Half Turn

FIGURE: fig5-point-symmetry-half-turn.png (full width)

After the $180°$ turn, where does vertex $A$ land? ______
Why does the outline look unchanged even though every vertex moved?

Fill in the blank. A regular $n$ -gon has point symmetry exactly when $n$ is ____________________ .

Does a regular pentagon have point symmetry? ______ Rule used: ____________________
Does a regular hexagon have point symmetry? ______ Why? ____________________

PAGE 9 — Practice · the four-way decision

Line, Point, Both, or Neither?

a) square ____________ b) rhombus (not square) ____________ c) isosceles trapezoid ____________ d) scalene triangle ____________
a) regular octagon ____________ b) regular heptagon ____________ c) kite ____________ d) circle ____________
Regular $12$ -gon: ______ lines · point symmetry? ______
Why does an equilateral triangle have three lines but no point symmetry?
Sketch a figure with point symmetry and zero lines of symmetry.
Sketch a figure with exactly one line of symmetry and no point symmetry.
Parallelogram (not a rectangle): line symmetry? ______ point symmetry? ______
Give a figure with line symmetry and no point symmetry. ____________

PAGE 10 — Apply and reason · Lesson 3.2

Apply It

Application. A playing card's design is unchanged when the card is turned upside down. Which kind of symmetry is that? Which does it not guarantee?
Application. A logo is four identical curved blades sweeping the same way around a center. Line symmetry? ______ Point symmetry? ______ Explain:
Error analysis. A student says every figure with point symmetry also has line symmetry. Counterexample:
Error analysis. A student says a regular polygon always has point symmetry. Counterexample and correct rule:
Reasoning. Why does the parity of $n$ decide point symmetry for a regular $n$ -gon?
Reasoning. A figure has exactly two perpendicular lines of symmetry. Why must it also have point symmetry?

PAGE 11 — Translations

3.3 Translations

FIGURE: fig6-translation-on-a-grid.png (full width)

Fill in the blank. A translation of $a$ units horizontally and $b$ units vertically is $(x, y) \rightarrow$ ( ____________ , ____________ ).

$A$ ______ $B$ ______ $C$ ______ · $A'$ ______ $B'$ ______ $C'$ ______
Translation rule shown: ____________________
Why are the dashed connecting segments parallel and congruent?

PAGE 12 — The four reflections

Four Mirror Lines

FIGURE: fig7-reflection-over-four-lines.png (full width)

Complete the frame.

Mirror line	Rule	What changes
$x$ -axis	$(x, y) \rightarrow$ ____________	____________
$y$ -axis	$(x, y) \rightarrow$ ____________	____________
$y = x$	$(x, y) \rightarrow$ ____________	____________
$y = -x$	$(x, y) \rightarrow$ ____________	____________

Rule over the $x$ -axis: ____________ over the $y$ -axis: ____________
Rule over $y = x$ : ____________ over $y = -x$ : ____________
In the $y = x$ panel, why is $A'$ in the same place as $A$ ?
Which named reflection swaps the coordinates and changes both signs? ____________

PAGE 13 — Practice · translate and reflect

Practice

FIGURE: fig12-blank-grids-for-transformations.png (full width)

Image under $(x, y) \rightarrow (x - 6,\ y + 2)$ : a) $(4, 1)$ → ______ b) $(-3, -5)$ → ______ c) $(0, 7)$ → ______
$D(1, 2)$ , $E(4, 2)$ , $F(4, 6)$ translated $3$ left and $5$ down: $D'$ ______ $E'$ ______ $F'$ ______ Rule: ____________
Image of $(-5, 2)$ over each line: $x$ -axis ______ $y$ -axis ______ $y = x$ ______ $y = -x$ ______
Reflect $J(2, 3)$ , $K(6, 3)$ , $L(6, 8)$ over the $x$ -axis, then over the $y$ -axis. Both images:
Image of $(-2, 6)$ under $(x, y) \rightarrow (x + 5,\ y - 5)$ : ______
Reflect $(3, -7)$ over the $x$ -axis ______ and over $y = x$ ______

PAGE 14 — Running a rule backwards

Finding a Preimage

A translation maps $(7, -2)$ to $(1, 3)$ . Rule: ____________ Image of $(0, 0)$ : ______
A reflection over the $y$ -axis produced $(-9, 4)$ . Preimage: ______
A translation $(x, y) \rightarrow (x + 2,\ y - 8)$ produced $(-1, -3)$ . Preimage: ______
A translation maps $(2, 2)$ to $(-4, 9)$ . Rule: ____________

Fill in the blank. To undo a translation, ____________________ what was added.

PAGE 15 — Apply and reason · Lesson 3.3

Apply It

Application. A game piece at $(3, 4)$ moves $7$ right and $2$ up each turn. Rule: ____________ After one turn: ______ After two: ______
Application. A design is flipped over the $y$ -axis for the other half of a logo. A point at $(5, -2)$ maps to ______
Error analysis. A student reflects $(4, 9)$ over $y = x$ and writes $(-9, -4)$ . Error and correct image:
Reasoning. Why does a reflection reverse orientation while a translation does not?
Reasoning. A point does not move under a reflection. What must be true about it? ____________

PAGE 16 — Rotations

3.4 Rotations

FIGURE: fig8-rotation-about-the-origin.png (full width)

Complete the frame.

Turn (counterclockwise)	Rule
$90°$	$(x, y) \rightarrow$ ____________
$180°$	$(x, y) \rightarrow$ ____________
$270°$	$(x, y) \rightarrow$ ____________

$90°$ rule: ____________
$180°$ rule: ____________ $270°$ rule: ____________
Which rotation is a half turn, and where else has it appeared in this chapter? ____________

Watch out. A $270°$ counterclockwise turn is the same as a ______° ____________ turn.

PAGE 17 — Dilations

Dilations from the Origin

FIGURE: fig9-dilation-from-the-origin.png (full width)

Fill in the blank. A dilation from the origin with scale factor $k$ is $(x, y) \rightarrow$ ( ______ , ______ ).

Scale factor in each panel: ______ and ______ Which is an enlargement? ____________
What do the dashed rays show?
Which transformation is not a rigid motion, and what does it change? ____________________

Complete the comparison table.

Transformation	Distances	Angle measures	Orientation
Translation	______	______	______
Reflection	______	______	______
Rotation	______	______	______
Dilation	______	______	______

PAGE 18 — Practice · rotate and dilate

Practice

FIGURE: fig12-blank-grids-for-transformations.png (full width)

$90°$ CCW: a) $(1, 4)$ → ______ b) $(-3, 2)$ → ______ c) $(0, -5)$ → ______
$180°$ : a) $(6, -1)$ → ______ b) $(-2, -8)$ → ______
$270°$ CCW: a) $(5, 3)$ → ______ b) $(-4, 7)$ → ______
$P(1, 1)$ , $Q(4, 1)$ , $R(4, 3)$ rotated $90°$ CCW: $P'$ ______ $Q'$ ______ $R'$ ______
Dilate $(8, -6)$ by $k = \tfrac12$ ______ and by $k = 3$ ______
A dilation maps $(2, 5)$ to $(10, 25)$ . $k$ = ______
$W(1,1)$ , $X(4,1)$ , $Y(4,3)$ , $Z(1,3)$ dilated by $k = 3$ : ______ Compare perimeters: ____________
$(-6, 2)$ rotated $90°$ CCW: ______
$(-6, 2)$ rotated $180°$ : ______
$(9, -12)$ dilated by $k = \tfrac13$ : ______
Which transformation preserves angle measures but not distances? ____________

PAGE 19 — Apply and reason · Lesson 3.4

Apply It

Application. A logo is enlarged from the origin by $2.5$ . A point at $(4, -2)$ maps to ______ What happens to the angles? ____________
Application. A robot arm at $(3, 0)$ swings $90°$ CCW about the origin → ______ $90°$ clockwise → ______
Error analysis. A student rotates $(5, 2)$ by $90°$ CCW and writes $(2, 5)$ . Error and correct image:
Error analysis. A student says $k = 3$ triples every angle measure. Correct it:
Reasoning. Why is the distance from the origin unchanged by any rotation, and how does that catch sign errors?
Reasoning. Why does a dilation produce a similar figure rather than a congruent one?

PAGE 20 — Naming a transformation

3.5 Reading the Picture Backwards

FIGURE: fig10-identify-the-transformation.png (full width)

The three questions, in order.

Is it the same ____________ ? If no → ____________________ .
Is the ____________________ reversed? If yes → ____________________ .
Did every point move the ____________________ ? If yes → ____________________ ; if no → ____________________ .

Which case is a dilation, and how do you know before computing? ____________________
Which case reverses orientation? ______
Which case has the same difference for every corresponding pair? ______
Name the transformation in Case C, and say how you separated it from a reflection.
Give the three questions in order.

PAGE 21 — Compositions

Two in a Row

FIGURE: fig11-composition-of-two-transformations.png (full width)

First transformation: ____________________ Second: ____________________
Why does the order matter?
Reflect $(2, 5)$ over the $x$ -axis, then translate $4$ right. Both images: ______ and ______
Same two in the opposite order, on $(2, 5)$ : ______ and ______ Compare: ____________
Reasoning. Two reflections over the $x$ -axis in a row. Equivalent single transformation? ____________
Describe a composition carrying $(1, 1)$ to $(-1, 5)$ , and state the order.

PAGE 22 — Practice · name it

Practice

Name the transformation.

a) $(3,1) \rightarrow (3,-1)$ , $(6,1) \rightarrow (6,-1)$ , $(6,4) \rightarrow (6,-4)$ ____________________

b) $(2,2) \rightarrow (7,5)$ , $(5,2) \rightarrow (10,5)$ , $(5,6) \rightarrow (10,9)$ ____________________

c) $(1,3) \rightarrow (2,6)$ , $(4,3) \rightarrow (8,6)$ , $(4,5) \rightarrow (8,10)$ ____________________

d) $(1,2) \rightarrow (-2,1)$ , $(4,2) \rightarrow (-2,4)$ , $(4,4) \rightarrow (-4,4)$ ____________________
Translation rule(s): ____________ Dilation $k$ : ______
Same side lengths, reversed orientation → ____________________ Ruled out: ____________________
Image half the size → ____________________ $k$ = ______
Same size, reversed orientation → ____________________
Every pair differs by $(-3, +2)$ → ____________________ Rule: ____________

PAGE 23 — Apply and reason · Lesson 3.5

Apply It

Application. A pattern reflects a shape over the $y$ -axis then translates it $6$ up. Final image of $(3, -1)$ : ______
Application. A photo is resized so $(6, 8)$ moves to $(9, 12)$ , centered at the origin. Transformation: ____________ $k$ = ______ Did the angles change? ______
Error analysis. A student sees the same size with reversed orientation and calls it a $180°$ rotation. Error:
Reasoning. Why check "same size?" before checking orientation?

PAGE 24 — Chapter review

Review

Review 1 (G.RLT.3 a, b). Lines of symmetry and point symmetry:

Figure	Lines	Point symmetry?
Square	______	______
Rhombus (not square)	______	______
Isosceles trapezoid	______	______
Regular heptagon	______	______
Regular decagon	______	______
Parallelogram (neither)	______	______

The two general rules for regular polygons: ____________________

Review 2 (G.RLT.3d). $\triangle ABC$ : $A(2,1)$ , $B(5,1)$ , $C(5,5)$ . Image under each, from the original each time:

Transformation	$A'$	$B'$	$C'$
$(x, y) \rightarrow (x - 6,\ y + 3)$	______	______	______
reflection over $y$ -axis	______	______	______
reflection over $y = -x$	______	______	______
rotation $90°$ CCW	______	______	______
rotation $270°$ CCW	______	______	______
dilation $k = 2$	______	______	______

Congruent to $\triangle ABC$ : ____________ Only similar: ____________ Why? ____________

Review 3 (G.RLT.3 c, d). $D(1,2)$ , $E(4,2)$ , $F(4,4)$ → $D'(-1,-2)$ , $E'(-4,-2)$ , $F'(-4,-4)$ .

Two corresponding side lengths: $DE$ = ______ , $D'E'$ = ______ ; $EF$ = ______ , $E'F'$ = ______
Single transformation: ____________________ Coordinate rule: ____________
Composition of two: ____________________ Order: ____________
How a reflection over a named line was ruled out: ____________________

Canva production notes

Page size: 8.5 × 11 in, 0.6 in margins. One workbook page per Canva page.
Type: page title H1 28 pt, section label H2 18 pt, body 12 pt, answer blanks 12 pt with a 1 pt rule.
Figures: place at the width noted beside each FIGURE: line. All figures are 200 dpi PNG on white.
Blank grids: fig12-blank-grids-for-transformations.png holds three empty coordinate grids and is used on pages 13, 18, and for any sketch item (11, 12, 18, 33, 34). Print it at full width so a student can plot a preimage and an image side by side.
Coordinate blanks are written as ______ pairs; keep them wide enough for a signed two-digit ordered pair.
Symbols: →, °, △, ½, and ⅓ must render in the body font; check the Canva font supports them before the first export.
Item numbers are continuous from 1 to 110 and must not be renumbered when pages are reordered.