Question

suppose line k has the equation x = 3. compare the areas of abcd and abcd=(t_(1,2)∘r_k)(abcd). suppose abcd is a rectangle with length ℓ and width w. its area is given by the expression. rigid motions sometimes preserve lengths and widths, so they change the value of the area. therefore, the areas of abcd and abcd be equal. similar reasoning applies (type an expression using ℓ and w as the variables.) never sometimes always

Explanation:

Step1: Recall area formula for rectangle

The area of a rectangle \(ABCD\) with length \(l\) and width \(w\) is \(A = lw\).

Step2: Understand rigid - motion properties

Rigid motions (translations, rotations, reflections) always preserve lengths and widths. Since area is calculated as the product of length and width (\(A = lw\)), and rigid - motions do not change the values of length and width, they do not change the area.

Step3: Analyze the transformation

The transformation \((T_{(1,2)}\circ r_{k})(ABCD)\) is a composition of a translation \(T_{(1,2)}\) (a translation by the vector \((1,2)\)) and a reflection \(r_{k}\) (a reflection over the line \(x = 3\)). Both translation and reflection are rigid motions.

Answer:

1. The area of rectangle \(ABCD\) is given by the expression \(lw\).
2. Rigid motions always preserve lengths and widths, so they never change the value of the area.
3. Therefore, the areas of \(ABCD\) and \(A''B''C''D''\) always be equal.