Ximpledu

Linear transformation

See how to find the points after a linear transformation
(translation/dilation/reflection/rotation matrix).
7 examples and their solutions.

Translation Matrix

Formula


xy + ab = x + ay + b

Example

△ABC is moved under the translation
(x, y) → (x + 4, y + 1).
Find the vertices of the moved triangle △A'B'C'.
A(2, 1), B(3, 4), C(5, 3)
Solution

Dilation Matrix

Formula


kI xy = x'y'
I: identity matrix

Example

△ABC is moved under the dilation of 2.
Find the vertices of the moved triangle △A'B'C'.
A(2, 1), B(3, 4), C(5, 3)
Solution

Reflection Matrix: x-axis

Formula


100-1 xy = x'y'

Example

△ABC is moved under the reflection in the x-axis.
Find the vertices of the moved triangle △A'B'C'.
A(2, 1), B(3, 4), C(5, 3)
Solution

Reflection Matrix: y-axis

Formula


-1001 xy = x'y'

Example

△ABC is moved under the reflection in the y-axis.
Find the vertices of the moved triangle △A'B'C'.
A(2, 1), B(3, 4), C(5, 3)
Solution

Reflection Matrix: Origin

Formula


-I xy = x'y'
I: identity matrix

Example

△ABC is moved under the reflection in the origin.
Find the vertices of the moved triangle △A'B'C'.
A(2, 1), B(3, 4), C(5, 3)
Solution

Reflection Matrix: y = x

Formula


0110 xy = x'y'

Example

△ABC is moved under the reflection in y = x.
Find the vertices of the moved triangle △A'B'C'.
A(2, 1), B(3, 4), C(5, 3)
Solution

Rotation Matrix

Formula


cos θ-sin θsin θcos θ xy = x'y'
Trigonometry (Right Triangle)

Example

△ABC is moved under the rotation of 60º counterclockwise about the origin.
Find the vertices of the moved triangle △A'B'C'.
A(2, 2), B(4, 2), C(4, 4)
Solution