Question

**#6.) error analysis
name the sequence of rigid motions that maps quadrilateral abcd onto efgh.
stiles says: \you would translate abcd to the right 5 and down

1. so t_{5,-2} to map abcd onto efgh.\

explain to stiles why he is incorrect.

Explanation:

Step1: Analyze the coordinates of corresponding points

Let's assume the coordinates of the points: For example, if \(A=(4,4)\), \(B=(4, - 1)\), \(C=(2,-1)\), \(D=(2,4)\) in quadrilateral \(ABCD\) and \(E=(4,-4)\), \(F=(4,-1)\), \(G=(2,-1)\), \(H=(2,-4)\) in quadrilateral \(EFGH\). A translation \(T_{5,-2}(x,y)=(x + 5,y-2)\) would not map \(ABCD\) to \(EFGH\) correctly.

Step2: Determine the correct rigid motion

First, reflect quadrilateral \(ABCD\) over the \(x -\)axis. The rule for reflection over the \(x -\)axis is \((x,y)\to(x,-y)\). Then translate the reflected figure. After reflection over the \(x -\)axis, if we then translate the reflected figure down by \(2\) units (translation rule \((x,y)\to(x,y - 2)\)).

Answer:

Stiles is incorrect because a translation \(T_{5,-2}\) is not the correct rigid - motion. The correct sequence of rigid motions is a reflection over the \(x -\)axis followed by a translation \(T_{0,-2}\) (or a translation \(T_{0,-2}\) followed by a reflection over the \(x -\)axis). A translation \(T_{5,-2}\) changes the \(x -\)coordinate in an incorrect way. For example, if we consider the \(x -\)coordinates of corresponding points (e.g., \(A\) and \(E\)), a translation \(T_{5,-2}\) would move a point \((x,y)\) to \((x + 5,y-2)\), but the \(x -\)coordinates of corresponding vertices of \(ABCD\) and \(EFGH\) (e.g., \(A=(4,4)\) and \(E=(4,-4)\)) have the same \(x -\)value. The key is the reflection (which changes the sign of the \(y -\)coordinate) and then a vertical translation.