Question

in exercises 1 - 3 perform the composition of transformations by graphing the preimage and image. also state whether the composition is a rigid or nonrigid motion.

1. rectangle defg with vertices d(-2, 7), e(2, 3), f(0, 1), and g(-4, 5):

a) translation along the rule (x, y)→(x + 6, y - 8)
b) reflection in the y - axis

Explanation:

Step1: Apply the translation

For point $D(-2,7)$: Using the translation rule $(x,y)\to(x + 6,y - 8)$, we have $x=-2,y = 7$. Then $x'=-2+6 = 4$ and $y'=7 - 8=-1$. So the translated - point $D_1(4,-1)$.
For point $E(2,3)$: $x = 2,y = 3$. Then $x'=2+6 = 8$ and $y'=3 - 8=-5$. So the translated - point $E_1(8,-5)$.
For point $F(0,1)$: $x = 0,y = 1$. Then $x'=0+6 = 6$ and $y'=1 - 8=-7$. So the translated - point $F_1(6,-7)$.
For point $G(-4,5)$: $x=-4,y = 5$. Then $x'=-4+6 = 2$ and $y'=5 - 8=-3$. So the translated - point $G_1(2,-3)$.

Step2: Apply the reflection in the y - axis

The rule for reflection in the y - axis is $(x,y)\to(-x,y)$.
For $D_1(4,-1)$: After reflection, $D'(-4,-1)$.
For $E_1(8,-5)$: After reflection, $E'(-8,-5)$.
For $F_1(6,-7)$: After reflection, $F'(-6,-7)$.
For $G_1(2,-3)$: After reflection, $G'(-2,-3)$.

Step3: Determine if it's rigid or non - rigid

A rigid motion preserves distance and angle measures. Both translation and reflection are rigid motions. So the composition of a translation and a reflection is a rigid motion.

Answer:

$D'(-4,-1)$; $E'(-8,-5)$; $F'(-6,-7)$; $G'(-2,-3)$; Rigid motion