Question

the figures shown are congruent. find a rigid motion that maps one figure to the other. give coordinate
notation for the transformations you use.

1. ( a b c d cong w x y z )

part 1: figure ( a b c d ) can be mapped onto ( w x y z ) by...
a rotation
b reflection
c translation
d the figures are not congruent
part 2: the coordinate notation is

Explanation:

Step1: Understand rigid motions

Rigid motions (rotations, reflections, translations) preserve the shape and size of a figure. Since \(ABCD\) and \(WXYZ\) are congruent (same shape and size), we check each transformation.

• Rotation: A rotation turns a figure around a point. But visually, \(ABCD\) and \(WXYZ\) do not seem to be related by a rotation.
• Reflection: A reflection flips a figure over a line. There is no obvious line of reflection that would map \(ABCD\) to \(WXYZ\).
• Translation: A translation slides a figure. If we look at the coordinates (assuming a standard grid - based coordinate system where each square is 1 unit):

Let’s assume \(A(x_1,y_1)\), \(B(x_2,y_2)\), \(C(x_3,y_3)\), \(D(x_4,y_4)\) and \(W(x_5,y_5)\), \(X(x_6,y_6)\), \(Z(x_7,y_7)\). If we consider the horizontal and vertical shifts. Suppose \(A\) is at \((- 6,2)\), \(B(-6, - 2)\), \(C(-2,-2)\), \(D(-2,2)\) and \(W(6,4)\), \(X(6, - 4)\), \(Z(2,-4)\), \(Y(2,4)\). The horizontal shift \(h=x_5 - x_1=6-(-6) = 12\) (right - ward) and vertical shift \(k = 4 - 2=2\) (up - ward). A translation \((x,y)\to(x + 12,y+2)\) maps \(ABCD\) to \(WXYZ\).

Step2: Coordinate notation

For coordinate notation, if we assume \(A(x_1,y_1)\), \(B(x_2,y_2)\), \(C(x_3,y_3)\), \(D(x_4,y_4)\) and \(W(x_5,y_5)\), \(X(x_6,y_6)\), \(Z(x_7,y_7)\), \(Y(x_8,y_8)\)
If \(A(-6,2)\), \(B(-6,-2)\), \(C(-2,-2)\), \(D(-2,2)\) and \(W(6,4)\), \(X(6,-4)\), \(Z(2,-4)\), \(Y(2,4)\)
The transformation rule is \((x,y)\to(x + 12,y + 2)\)

Answer:

1. \(ABCD\cong WXYZ\) by \(C\) (Translation).

Part 2: If we assume \(A(-6,2)\), then \(W\) has coordinates \((6,4)\) (using the translation rule \((x,y)\to(x + 12,y+2)\))