Standards: G-CO.A.4 · G-CO.A.5
A locksmith cuts a spare key and hands it back the wrong way round. It still fits — the teeth are in the same places, the key has just been turned.
A different shop hands back a key that looks right until you try it. Same shape, mirrored. That one will never work.
Both are rigid — nothing stretched, nothing shrank. But one is a turn and the other is a flip, and only one of them still opens the door.
Which of the two could you fix by turning it over?
Given a point or figure and a rotation angle about the origin, you will produce the image and check it by applying the inverse rotation.
Definitions and rules
Rotation about the origin
All counter-clockwise, all about the origin.
Why it has to be this: a quarter turn sends the horizontal axis onto the vertical one, so x and y swap roles and one of them changes sign.
Why 180° is the one to remember
R180 just negates both coordinates — no swapping. It is the easiest to recall and the easiest to check.
And 90° applied three times equals 270°, so if you remember only the quarter turn you can always get the rest.
Worked examples
Example 1 — a whole triangle through 270°
Rotate triangle A(2, −1), B(4, 2), C(−1, 3) by 270° counterclockwise about the origin.
Suggested work
Givens
Tools
Primitives
Computation
Notice: Each ordered pair swaps roles, then the original x changes sign.
Why the rule has this form
A 270° counterclockwise turn is a 90° clockwise turn. The axes exchange roles, and the new y points opposite the original x direction.
Example 2 — recovering a preimage from its rotated image
P′(−5, 2) is the image of P after a 90° counterclockwise rotation about the origin. Find P.
Suggested work
Givens
Tools
Primitives
Computation
Notice: Undo a rotation with the turn that completes the full 360° cycle.
Why the inverse check works
R90 followed by R270 totals 360°, which returns every point to its starting location.
Example 3 — a quarter turn
Rotate P(3, 4) by 90° counter-clockwise about the origin.
Suggested work
Givens
Tools
Primitives
Computation
Notice: the coordinates swapped places, then one changed sign.
Common traps
Clockwise 90° is R270
The label R90 in this book means 90° counterclockwise. A 90° clockwise turn follows R270(x, y) = (y, −x). The broken assumption is treating the angle size as enough information while ignoring direction.
Practice
Rotate R(4, −7) by 180° about the origin.
Answer — tap to show
R′(−4, 7)
Suggested work — tap to show
Givens
R(4, −7) 180° about the originTools
R180(x, y) = (−x, −y)Primitives
x = 4 y = −7Computation
R180(4, −7) = (−(4), −(−7)) = (−4, 7)Rotate Q(3, 2) by 270° counterclockwise about the origin.
Answer — tap to show
Q′(2, −3).
Rotate Q(−2, 5) by 90° counter-clockwise about the origin.
Answer — tap to show
Q′(−5, −2)
Suggested work — tap to show
Givens
Q(−2, 5) 90° CCW about the originTools
R90(x, y) = (−y, x)Primitives
x = −2 y = 5Computation
R90(−2, 5) = (−(5), (−2)) = (−5, −2)Rotate segment MN with endpoints M(−2, 1) and N(3, 1) by 90° counterclockwise about the origin.
Answer — tap to show
M′(−1, −2) and N′(−1, 3).
Suggested work — tap to show
Givens
M(−2, 1) N(3, 1) R90Tools
R90(x, y) = (−y, x)Primitives
Apply the rule to both endpoints.Computation
M(−2, 1) → M′(−1, −2) N(3, 1) → N′(−1, 3)A classmate is asked to rotate S(4, 1) by 90° clockwise and writes S′(−1, 4). Diagnose the error and give the correct image.
Answer — tap to show
They used the counterclockwise R90 rule. A 90° clockwise turn uses R270, so S′(1, −4).
Suggested work — tap to show
Givens
S(4, 1) 90° clockwise classmate wrote S′(−1, 4)Tools
90° clockwise = R270 R270(x, y) = (y, −x)Primitives
x = 4 y = 1Computation
R270(4, 1) = (1, −4) Their result, (−1, 4), comes from R90(4, 1).