Question

suppose line k has the equation x = 3. compare the areas of abcd and abcd (t(1,2) o k)(abcd). suppose abcd is a rectangle with length l and width w. its area is given by the expression. rigid motions preserve lengths and widths, so they change the value of the area. therefore, the areas of abcd and abcd be equal. similar reasoning applies when abcd is not a rectangle. (type an expression using l and w as the variables.)

Explanation:

Step1: Recall area formula for rectangle

The area formula for a rectangle is $A = l\times w$, where $l$ is the length and $w$ is the width.

Step2: Apply to rectangle ABCD

For rectangle $ABCD$ with length $l$ and width $w$, its area $A_{ABCD}=l\times w$.

Step3: Consider rigid - motion property

Rigid motions (translations, rotations, reflections) preserve lengths and widths. So if $A'B'C'D'$ is the result of a rigid - motion of $ABCD$, the lengths and widths of $A'B'C'D'$ are the same as those of $ABCD$.

Step4: Find area of $A'B'C'D'$

The area of $A'B'C'D'$, $A_{A'B'C'D'}=l\times w$ (since lengths and widths are preserved).

Answer:

The area of rectangle $ABCD$ is given by the expression $l\times w$, and the area of $A'B'C'D'$ (the result of a rigid - motion of $ABCD$) is also $l\times w$.