Question

1. triangle abc has vertices at a(3, - 3), b(8, - 3), and c(8, - 6) and triangle def has vertices at d(- 3, 3), e(2, 3), and f(2, 6).

give a sequence of transformations that would map abc onto def.

Explanation:

Step1: Analyze the x - coordinate change

For point \(A(3,-3)\) and \(D(-3,3)\), the x - coordinate changes from \(x = 3\) to \(x=-3\). The transformation for the x - coordinate is \(x\to x - 6\) (since \(3-6=-3\)). The y - coordinate changes from \(y=-3\) to \(y = 3\), which is \(y\to - y\).
First, perform a translation. Translate triangle \(ABC\) 6 units to the left. The rule for translation is \((x,y)\to(x - 6,y)\).
After translation:

• \(A(3,-3)\to A'(3 - 6,-3)=(-3,-3)\)
• \(B(8,-3)\to B'(8 - 6,-3)=(2,-3)\)
• \(C(8,-6)\to C'(8 - 6,-6)=(2,-6)\)

Step2: Analyze the y - coordinate change

Now, for the translated points \(A'(-3,-3)\), \(B'(2,-3)\), \(C'(2,-6)\) and the target points \(D(-3,3)\), \(E(2,3)\), \(F(2,6)\). The transformation for the y - coordinate is \(y\to - y\).
Reflect the translated triangle (after translation 6 units left) over the x - axis. The rule for reflection over the x - axis is \((x,y)\to(x,-y)\).
For \(A'(-3,-3)\), \((-3,-(-3))=(-3,3)=D\)
For \(B'(2,-3)\), \((2,-(-3))=(2,3)=E\)
For \(C'(2,-6)\), \((2,-(-6))=(2,6)=F\)

Answer:

First, translate triangle \(ABC\) 6 units to the left. Then, reflect the translated triangle over the \(x\) - axis.