SME
Grade 10 Math · Problem type
Symmetry of a rectangle
Use this for “Which rotations carry a 6 by 4 rectangle onto itself? How many lines of symmetry does it have?”
Diagram
Rotate 180° counterclockwise about (4, 3).
A(1, 1) → A′(7, 5); B′(1, 5), C′(1, 1), D′(7, 1).
The image has the same side lengths and angles.
2 lines of symmetry.
Rotational symmetry of order 2: turns of 180° about the center carry it onto itself.
Point symmetry about its center (4, 3): every corner has a partner straight across it, the same distance away, so a half turn swaps them.
This is an example. Type your own values to start a new problem.
w
Width
example
h
Height
example
t
Turn
example · °
L
Lines of symmetry
worked out (not typed)
n
Order of rotational symmetry
worked out (not typed)
f
Carried onto itself
worked out (not typed)
Formulas
L
lines
when
the
sides
are
w
and
h
2 lines when the sides are 6 and 4
order
n
when
the
sides
are
w
and
h
order 2 when the sides are 6 and 4
f
:
does
a
turn
of
t
carry
it
onto
itself,
order
n
180
360 ÷ 2
is
a
whole
number,
so
yes
Assumptions
•
The rectangle is w squares wide and h tall, and it turns about its center.
•
A rectangle that isn’t a square has 2 lines of symmetry and is carried onto itself by 180° and 360° turns.
•
A square (w = h) adds 90° and 270° turns and its two diagonals: 4 lines.
Step-by-step
WE KNOW
w = 6, h = 4, t = 180°
FIND
lines of symmetry (L), order of rotational symmetry (n), carried onto itself (f)
Step 1 · Find lines of symmetry (L)
Use L lines when the sides are w and h
Not a square: only the two midlines; a diagonal flips it onto a different rectangle.
L
=
2
(
w
=
6
and
h
=
4:
w
≠
h
,
not
a
square)
Step 2 · Find order of rotational symmetry (n)
Use order n when the sides are w and h
A rectangle lands on itself every half turn.
n
=
2
(
w
=
6
and
h
=
4:
w
≠
h
,
not
a
square)
Step 3 · Find carried onto itself (f)
Use f: does a turn of t carry it onto itself, order n
The
turn
is
a
whole
number
of
360°
÷
n
steps:
the
figure
lands
on
itself.
180
360 ÷ 2
is
a
whole
number:
yes
f
=
does
a
turn
of
180
carry
it
onto
itself,
order
2
f
=
yes
Check
Put the numbers back in. Both sides should match.
2 = 2 ✓
2 = 2 ✓
180
360 ÷ 2
is
a
whole
number,
so
yes
✓
Topic check
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Related lessons
Rigid motions, compositions and symmetry
Main lesson
A glide reflection
Use this for “Translate A(1, 2) 5 units right, then reflect it across the x-axis.”
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Use this for “Rotate A(6, 2) 90° counterclockwise about the point (1, 1).”
Reflecting across y = x or y = −x
Use this for “Reflect A(4, 1) across the line y = x” or “across y = −x.”
Which motion is the rule?
Use this for “Is (x, y) → (2x, 2y) a rigid motion? Which move is (x, y) → (−y, x)?”