SME
Grade 10 Math · Geometry
Rigid motions, compositions and symmetry
Use this for “Reflect A(2, 5) across the y-axis, then rotate the image 90° about the origin. Where does A end up?”
Diagram
Reflect across the y-axis, then rotate 90° counterclockwise about (0, 0).
A(2, 5) → A′(−2, 5) → A″(−5, −2).
Both moves keep lengths and angles, so the final image is congruent to the figure.
This is an example. Type your own values to start a new problem.
A(
x
,
y
)
→
A′(
x′
,
y′
)
→
A″(
x″
,
y″
)
Formulas
x
′
=
−
x
−2 = −2
y
′
=
y
5 = 5
x
″
=
−
y
′
−5 = −5
y
″
=
x
′
−2 = −2
Assumptions
•
First reflect across the y-axis: (x, y) → (−x, y). Then rotate 90° counterclockwise about the origin: (x, y) → (−y, x).
•
Each move keeps lengths and angles, so the final image is congruent to the figure.
•
Order matters: swapping the two moves can land somewhere else.
Step-by-step
WE KNOW
x = 2, y = 5
FIND
x of A′ (x′), y of A′ (y′), x of A″ (x″), y of A″ (y″)
Step 1 · Find x of A′ (x′)
Use x′ = −x
Reflecting
across
the
y
-axis
changes
the
sign
of
x
.
x
′
=
−2
Step 2 · Find y of A′ (y′)
Use y′ = y
Reflecting
across
the
y
-axis
keeps
y
.
y
′
=
5
Step 3 · Find x of A″ (x″)
Use x″ = −y′
A
quarter
turn
counterclockwise:
the
new
x
is
the
old
y
with
its
sign
changed.
x
″
=
−5
Step 4 · Find y of A″ (y″)
Use y″ = x′
A
quarter
turn
counterclockwise:
the
new
y
is
the
old
x
.
y
″
=
−2
Check
Put the numbers back in. Both sides should match.
−2 = −2 ✓
5 = 5 ✓
−5 = −5 ✓
−2 = −2 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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Use this for “Reflect A(4, 1) across the line y = x” or “across y = −x.”
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