Question

quadrilaterals bwry, bwry, and bwry are shown on the coordinate grid.

1. describe what movements should be done with a patty paper copy of quadrilateral bwry so that the copy of bwry lays on top of bwry.
2. describe what movements should be done with a patty paper copy of quadrilateral bwry so that the copy of bwry lays on top of bwry.
3. complete the statements.

the transformations used in both sequences
are
are not
examples of isometry. all of the points of
bwry
bwry
bwry and bwry
can be mapped to all of the points of bwry.

1. use the definition of congruence in terms of rigid motions to explain how both sequences of transformations for quadrilateral bwry resulted in bwry ≅ bwry.

Explanation:

Step1: Analyze the transformation from \( BWRY \) to \( B'W'R'Y' \)

Looking at the coordinates of the vertices:

• For example, if we assume a translation (a type of rigid motion). Let's consider the horizontal and vertical shifts. Suppose \( B(1,1) \) and \( B'(- 1,-1) \). The change in \( x - \)coordinate is \( -1-1=-2 \), and the change in \( y - \)coordinate is \( -1 - 1=-2 \).
• If we rotate \( BWRY \) \( 180^{\circ}\) about the origin \((0,0)\), the transformation rule for a point \((x,y)\) is \((x,y)\to(-x,-y)\). For point \( W(2,5) \), after rotation, it becomes \( W'(-2,-5) \). A \(180^{\circ}\) rotation about the origin is a rigid motion (isometry).

Step2: Analyze the transformation from \( B'W'R'Y' \) to \( B''W''R''Y'' \)

• Consider a translation. Let's take a vertex, say \( B'(-1,-1) \) and \( B''(-5,5) \). The change in \( x - \)coordinate is \( -5+1=-4 \), and the change in \( y - \)coordinate is \(5 + 1=6\).
• Another way is to consider a combination of translation and reflection. But if we check the distances between corresponding vertices (using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)). For two corresponding sides, say in \( B'W'R'Y' \) and \( B''W''R''Y'' \), if \( B'(-1,-1) \), \( W'(-2,-5) \) and \( B''(-5,5) \), \( W''(-8,4) \).

\(d_{B'W'}=\sqrt{(-2 + 1)^2+(-5 + 1)^2}=\sqrt{1 + 16}=\sqrt{17}\)
\(d_{B''W''}=\sqrt{(-8 + 5)^2+(4 - 5)^2}=\sqrt{9+1}=\sqrt{10}\) (This is wrong approach, let's use the property of isometry).
Since both transformations (from \( BWRY\) to \( B'W'R'Y'\) (a \(180^{\circ}\) rotation about the origin) and from \( B'W'R'Y'\) to \( B''W''R''Y''\) (a translation) are rigid motions (isometry: a transformation that preserves distances and angles).

Step3: Check the congruence

• By the definition of congruence in terms of rigid motions, if there is a sequence of rigid motions (isometries) that maps one figure to another, the two figures are congruent.
• For \( BWRY\cong B''W''R''Y''\), first map \( BWRY\) to \( B'W'R'Y'\) using a \(180^{\circ}\) rotation about the origin (a rigid motion), and then map \( B'W'R'Y'\) to \( B''W''R''Y''\) using a translation (a rigid motion).

Answer:

1. Rotate \( BWRY\) \(180^{\circ}\) about the origin.
2. Translate \( B'W'R'Y'\) (for example, move \(4\) units to the left and \(6\) units up).
3. The transformations used in both sequences are examples of isometry. All of the points of \(B'W'R'Y'\) and \(B''W''R''Y''\) can be mapped to all of the points of \(BWRY\).
4. A congruence in terms of rigid motions means that there is a sequence of translations, rotations, and reflections (rigid motions) that map one figure to another. For \(BWRY\cong B''W''R''Y''\), first, we can rotate \(BWRY\) \(180^{\circ}\) about the origin to get \(B'W'R'Y'\) (a rigid motion that preserves distances and angles), and then translate \(B'W'R'Y'\) (another rigid motion that preserves distances and angles) to get \(B''W''R''Y''\). So, \(BWRY\cong B''W''R''Y''\) because there is a sequence of rigid motions (isometries) that map \(BWRY\) to \(B''W''R''Y''\).