Question

1. rectangle abcd with vertices a(1, 0), b(7, 2), c(8, -1), and d(2, -3): (a) translation: (x, y)→(x - 8, y - 3) (b) reflection: in the x - axis a(, ) b(, ) c(, ) d(, )

Explanation:

Step1: Apply translation rule

For point \(A(1,0)\), using the translation \((x,y)\to(x - 8,y - 3)\), we have \(x=1,y = 0\). Then \(x'=1-8=-7\) and \(y'=0 - 3=-3\), so \(A'(-7,-3)\).
For point \(B(7,2)\), \(x = 7,y=2\). Then \(x'=7-8=-1\) and \(y'=2 - 3=-1\), so \(B'(-1,-1)\).
For point \(C(8,-1)\), \(x = 8,y=-1\). Then \(x'=8-8 = 0\) and \(y'=-1-3=-4\), so \(C'(0,-4)\).
For point \(D(2,-3)\), \(x = 2,y=-3\). Then \(x'=2-8=-6\) and \(y'=-3-3=-6\), so \(D'(-6,-6)\).

Step2: Apply reflection rule

The rule for reflection in the \(x -\)axis is \((x,y)\to(x,-y)\).
For \(A'(-7,-3)\), after reflection, \(A''(-7,3)\).
For \(B'(-1,-1)\), after reflection, \(B''(-1,1)\).
For \(C'(0,-4)\), after reflection, \(C''(0,4)\).
For \(D'(-6,-6)\), after reflection, \(D''(-6,6)\).

Answer:

\(A'(-7,3)\)
\(B'(-1,1)\)
\(C'(0,4)\)
\(D'(-6,6)\)