Question

suppose line k has the equation x = 3. compare the areas of abcd and abcd=(t(r,2)∘r1)(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 of a rectangle with length $l$ and width $w$ is $A = lw$.

Step2: Understand rigid - motion properties

Rigid motions (translations, rotations, reflections) preserve lengths and widths. Since area is calculated based on lengths and widths, rigid motions do not change the area.

Step3: Compare areas

Since $(T_{(r_1,r_2)}\circ r_{l})(ABCD)$ is a rigid - motion transformation of $ABCD$, the areas of $ABCD$ and $A''B''C''D''$ are equal.

Answer:

The area of rectangle $ABCD$ is $lw$. Rigid motions do not change the value of the area. The areas of $ABCD$ and $A''B''C''D''$ are equal.