Question

check your understanding
examples 1, 2 identify the type of rigid transformation shown as a reflection, translation, or rotation.
1.
2.

1. △xyz with vertices x(−9,4), y(−9,−3), z(−1,−3)

find the coordinates of the image after each rigid transformation.
a. reflection in the x - axis
b. translation along (4,6)
c. counterclockwise rotation of 270° about the origin

1. parallelogram pqrs with vertices p(−2,5), q(−9,5), r(−9,−1), s(−2,−1)

a. reflection in the y - axis
b. translation along −3,1
c. rotation of 180° about the origin
guided practice

1. consider △xyz with vertices x(−4,2), y(−1,1), and z(2,3). what are the coordinates of the vertices of its image after a rotation 90° counterclockwise about the origin?

watch out!
verify vertices. some transformations may look equivalent but are not. the transformation on the plane of fghj may look like a translation, but the vertices are in different positions. be sure to verify that corresponding vertices are labeled correctly.
a rotation is a function that moves every point of a pre - image and direction about a fixed point called the center of rotation and origin as the center when you perform a rotation.
90° rotation counterclockwise
(x,y)→(−y,x)
180° rotation
(x,y)→(−x,−y)
270°
example 5 rotations on the coordinate plane
parallelogram fghj has vertices f(2,1), g(7,1), h(6,−3), and j(1,−3).
what are the coordinates of the vertices of its image after a rotation 180° about the origin?
the figure is in quadrants i and iv. its image will be oriented differently and it will be in quadrants ii and iii.
f(2,1)→f(−2,−1)
g(7,1)→g(−7,−1)
h(6,−3)→h(−6,3)
j(1,−3)→j(−1,3)
transformations in the plane

Explanation:

Step1: Recall reflection rule for x - axis

For a reflection in the x - axis, the rule is $(x,y)\to(x, - y)$.
For $\triangle XYZ$ with $X(-9,4)$, $Y(-9, - 3)$, $Z(-1,-3)$:
$X(-9,4)\to X'(-9,-4)$
$Y(-9,-3)\to Y'(-9,3)$
$Z(-1,-3)\to Z'(-1,3)$

Step2: Recall translation rule

For a translation along the vector $(4,6)$, the rule is $(x,y)\to(x + 4,y + 6)$.
$X(-9,4)\to X''(-9 + 4,4+6)=(-5,10)$
$Y(-9,-3)\to Y''(-9 + 4,-3 + 6)=(-5,3)$
$Z(-1,-3)\to Z''(-1+4,-3 + 6)=(3,3)$

Step3: Recall 270 - degree counter - clockwise rotation rule

A 270 - degree counter - clockwise rotation about the origin has the rule $(x,y)\to(y,-x)$.
$X(-9,4)\to X'''(4,9)$
$Y(-9,-3)\to Y'''(-3,9)$
$Z(-1,-3)\to Z'''(-3,1)$

Answer:

a. $X'(-9,-4),Y'(-9,3),Z'(-1,3)$
b. $X''(-5,10),Y''(-5,3),Z''(3,3)$
c. $X'''(4,9),Y'''(-3,9),Z'''(-3,1)$