SME
Grade 8 Math · Problem type
Rotation about the origin
Use this for “Rotate (3, 1) by 90° counterclockwise about (0, 0).”
Diagram
Rotate 90° counterclockwise about (0, 0): (x, y) → (−y, x).
A(3, 1) → A′(−1, 3); B′(−1, 6), C′(−4, 3).
The image has the same side lengths and angles.
This is an example. Type your own values to start a new problem.
x
x of A
example
y
y of A
example
r
Angle
example · °
x′
x of A′
worked out (not typed)
y′
y of A′
worked out (not typed)
Formulas
x
′
=
x
-coordinate
of
(
x
,
y
)
turned
r
°
(x
of
A′
=
x-coordinate
of
(x
of
A,
y
of
A)
turned
angle°)
−1
=
x
-coordinate
of
(3,
1)
turned
90°
y
′
=
y
-coordinate
of
(
x
,
y
)
turned
r
°
(y
of
A′
=
y-coordinate
of
(x
of
A,
y
of
A)
turned
angle°)
3
=
y
-coordinate
of
(3,
1)
turned
90°
Assumptions
•
A rotation turns the figure about the center; a positive angle turns counterclockwise.
•
A quarter turn swaps the coordinates and changes one sign: (x, y) → (−y, x).
•
A half turn changes both signs: (x, y) → (−x, −y).
Step-by-step
WE KNOW
x = 3 (x of A), y = 1 (y of A), r = 90° (angle)
FIND
x of A′ (x′), y of A′ (y′)
Step 1 · Find x of A′ (x′)
Use x′ = x-coordinate of (x, y) turned r° (x of A′ = x-coordinate of (x of A, y of A) turned angle°)
A
quarter
turn
counterclockwise:
(
x
,
y
)
→
(−
y
,
x
).
(3, 1) → (−1, 3)
x
′
=
−1
Step 2 · Find y of A′ (y′)
Use y′ = y-coordinate of (x, y) turned r° (y of A′ = y-coordinate of (x of A, y of A) turned angle°)
A
quarter
turn
counterclockwise:
(
x
,
y
)
→
(−
y
,
x
).
y
′
=
3
Check
Put the numbers back in. Both sides should match.
−1 = −1 ✓
3 = 3 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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Related lessons
Translations, rotations, reflections and dilations
Main lesson
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