Reflections in the Coordinate Plane
4.9 Rigid Motions and Congrunce
Quadrilateral ABCD has vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3). Graph ABCD and its image under reflection in the x‐axis. Compare the coordinates of each vertex with the coordinates of its image.
**Use the vertical grid lines to find the corresponding point for each vertex so that the x‐axis is equidistant from each vertex and its image.
I CAN...
Write Motion Rules for Rigid Motion in the Coordinate Plane A(1,1) ⇒ A'( , )
B(3,2) ⇒ B'( , )
C(4,­1) ⇒ C'( , )
D(2,­3) ⇒ D'( , )
What conclusion can you make about the relationship of the x­value and the y­value?
(x, y) ⇒( x, ­y)
Jul 30­10:36 AM
Jul 30­10:36 AM
Reflections in the Coordinate Plane
Quadrilateral ABCD has vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3). Graph ABCD and its image under reflection in the y‐axis. Compare the coordinates of each vertex with the coordinates of its image.
Use the horizontal grid lines to find the corresponding point for each vertex so that the y‐axis is equidistant from each vertex and its image.
Reflections in the Coordinate Plane
Quadrilateral ABCD has vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3). Graph ABCD and its image under reflection in the line y = x. Compare the coordinates of each vertex with the coordinates of its image.
A(1,1) ⇒ A'( , )
B(3,2) ⇒ B'( , )
C(4,­1) ⇒ C'( , )
D(2,­3) ⇒ D'( , )
A(1, 1) ⇒ A'( , )
B(3, 2) ⇒ B'( , )
C(4,­1) ⇒ C'( , )
D(2,­3) ⇒ D'( , )
What conclusion can you make about the relationship of the x­value and the y­value?
What conclusion can you make about the relationship of the x­value and the y­value?
(x, y) ⇒( , )
(x, y) ⇒( , )
Jul 30­10:36 AM
Jul 30­10:36 AM
Translations in the Coordinate Plane
Motion Rules for Reflections in the Coordinate Plane
Reflection x­axis
y­axis
origin
y=x
preimage to image
(x, y) ⇒(x, ­y) (x, y) ⇒(­x, y) (x, y) ⇒(­x,­y) (x, y) ⇒(y, x) How to find coordinates Multiply the y­coordinate by ­1. Multiply the x­coordinate by ­1. Multiply both coordinates by ­1. Interchange the x­ and y­coordinates. Example (2, 3) ⇒(2, ­3) (2, 3) ⇒(­2, 3) (2, 3) ⇒(­2, ­3) (2, 3) ⇒(3, 2)
Jul 30­10:36 AM
Triangle TUV has vertices T(–1, –4), U(6, 2), and V(5, –5). If ΔTUV is translated 3 units up and 5 units left to create ΔT'U'V', what are the coordinates of the vertices of ΔT'U'V'? You are asked to find the coordinates of the image of ΔTUV after a translation of 3 units up and 5 units left which can be identified as vector <‐5, 3>
(x, y) → (x – 5, y + 3).
Jul 30­10:36 AM
1
Translations in the Coordinate Plane
Find the translation that moves AB with endpoints A(2, 4) and B(–1, –3) to A'B' with endpoints A'(5, 2) and B'(2, –5).
Rotations in the Coordinate Plane
Find the coordinates of trapezoid LMNO with vertices L(2, 1), M(5, 1), N(1, –5), and O(0, –2) rotated 90 clockwise.
Jul 30­10:36 AM
Jul 30­10:36 AM
Rotations in the Coordinate Plane
Triangle TUV has vertices T(–1, –4), U(6, 2), and V(5, –5). If ΔTUV is rotated 180, what are the coordinates of the vertices of ΔT'U'V'? Motion Rules for Rotations in the Coordinate Plane
Rotation 90o clockwise
90o counter­
clockwise
180o preimage to image
(x, y) ⇒(y, ­x) (x, y) ⇒(­y, x) (x, y) ⇒(­x,­y) How to find coordinates Example Jul 30­10:36 AM
Swap the x­ and y­ Swap the x­ and y­ coordinate and coordinate and multiply multiply the the x­coordinate by ­1. y­coordinate by ­1. (2, 3) ⇒(3, ­2) (2, 3) ⇒(­3, 2) Multiply both coordinates by ­1. (2, 3) ⇒(­2, ­3) Jul 30­10:36 AM
Composite Transformations in the Coordinate Plane
Composite transformations: combination of 2 transformations
Glide reflection: composition of a translation followed by a
reflection.
ΔABC has vertices A(-5,-1), B(-3,-5), C(-1,-1).
Translate along <0,4> and reflect in the y-axis.
Motion Rule:
(x,y)⇒(x,y+4)⇒(-x, y)
A(­5, ­1)⇒A'(
,
Δ ABC has vertices A(-5,-1), B(-6,-4), C(-2,-4).
Translate along <-3,-3> and a reflection in the x-axis.
Motion Rule:
(x,y)⇒(x+3,y-2)⇒(x, -y)
)⇒A''(
B(­3, ­5)⇒B'( , )⇒B''(
C(­1, ­1)⇒C'( , )⇒C''(
Composite Transformations in the Coordinate Plane
,
,
)
)
,
)
Jul 30­10:36 AM
A(­5, ­1)⇒A'(
,
)⇒A''(
B(­6, ­4)⇒B'(
,
)⇒B''(
,
)
C(­2, ­4)⇒C'(
,
)⇒C''(
,
)
,
)
Jul 30­10:36 AM
2
Composite Transformations in the Coordinate Plane
Δ ABC has vertices A(-4,5), B(3,2), C(1,5).
Translate along <-2,-5> and rotate 90o clockwise.
Motion Rule:
(x,y)⇒(x-2,y-5)⇒(y, -x)
A(­4, 5)⇒A'(
B(3, 2)⇒B'(
C(1, 5)⇒C'(
,
,
,
)⇒A''(
)⇒B''(
)⇒C''(
, )
, )
, )
Jul 30­10:36 AM
3