Question

1. zoe wants to move her garden, and she needs to figure out what the points will be on the graph after she translates it. her garden is a rectangle and she is translating it 6 units up and 7 units left. the vertices for the rectangle abcd are (2,3), (11,3), (2,-1) & (11,-1). a. what are the coordinates of a, b, c, and d? b. how does the length of ca compare to the length of line ca? c. how does the area of rectangle abcd compare to the area of rectangle abcd?

Explanation:

Step1: Apply translation rule

The translation rule is $(x,y)\to(x - 7,y + 6)$. For point $A(2,3)$: $A'=(2-7,3 + 6)=(-5,9)$; for point $B(11,3)$: $B'=(11-7,3 + 6)=(4,9)$; for point $C(2,-1)$: $C'=(2-7,-1 + 6)=(-5,5)$; for point $D(11,-1)$: $D'=(11-7,-1 + 6)=(4,5)$.

Step2: Calculate length of CA and C'A'

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For $C(2,-1)$ and $A(2,3)$, $x_1 = 2,y_1=-1,x_2 = 2,y_2 = 3$. Then $CA=\sqrt{(2 - 2)^2+(3+1)^2}=4$. For $C'(-5,5)$ and $A'(-5,9)$, $x_1=-5,y_1 = 5,x_2=-5,y_2 = 9$. Then $C'A'=\sqrt{(-5 + 5)^2+(9 - 5)^2}=4$. So the lengths are equal.

Step3: Calculate areas

The area of a rectangle is $A = l\times w$. For rectangle $ABCD$, length $l=11 - 2=9$ and width $w=3+1 = 4$, so $A_{ABCD}=9\times4 = 36$. For rectangle $A'B'C'D'$, length $l'=4+5 = 9$ and width $w'=9 - 5=4$, so $A_{A'B'C'D'}=9\times4=36$. The areas are equal.

Answer:

a. $A'(-5,9),B'(4,9),C'(-5,5),D'(4,5)$
b. The lengths of $CA$ and $C'A'$ are equal.
c. The areas of rectangle $ABCD$ and rectangle $A'B'C'D'$ are equal.