SME
Grade 10 Math · Problem type
Rotating about a point
Use this for “Rotate A(6, 2) 90° counterclockwise about the point (1, 1).”
Diagram
Rotate 90° counterclockwise about (1, 1).
A(6, 2) → A′(0, 6); B′(1, 7), C′(−1, 7).
The image has the same side lengths and angles.
This is an example. Type your own values to start a new problem.
A(
x
,
y
)
→
A′(
x′
,
y′
)
a
x of the center
example
b
y of the center
example
Formulas
x
′
=
a
−
(
y
−
b
)
0 = 1 − (2 − 1)
y
′
=
b
+
(
x
−
a
)
6 = 1 + (6 − 1)
Assumptions
•
The turn is 90° counterclockwise about the center (a, b).
•
Measure A from the center, turn that step a quarter turn, and add it back to the center.
•
So x′ = a − (y − b) and y′ = b + (x − a).
Step-by-step
WE KNOW
x = 6, y = 2, a = 1, b = 1
FIND
x of A′ (x′), y of A′ (y′)
Step 1 · Find x of A′ (x′)
Use x′ = a − (y − b)
A quarter turn sends the step up from the center to a step left.
x
′
=
1
−
(2
−
1)
x
′
=
0
Step 2 · Find y of A′ (y′)
Use y′ = b + (x − a)
A quarter turn sends the step right from the center to a step up.
y
′
=
1
+
(6
−
1)
y
′
=
6
Check
Put the numbers back in. Both sides should match.
0 = 1 − (2 − 1) ✓
6 = 1 + (6 − 1) ✓
Topic check
See what you know: 5 questions on this topic
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Related lessons
Rigid motions, compositions and symmetry
Main lesson
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