Question

1. rotate rectangle abcd 90° counterclockwise. then, translate it along the vector <3, 4>.

Explanation:

Step1: Identify original coordinates

Assume \(A(-4,4)\), \(B(-2,4)\), \(C(-2,1)\), \(D(-4,1)\)

Step2: Apply 90 - degree counter - clockwise rotation formula

The rotation formula for a 90 - degree counter - clockwise rotation about the origin \((x,y)\to(-y,x)\).
For \(A(-4,4)\), \(A'(-4, - 4)\); for \(B(-2,4)\), \(B'(-4,-2)\); for \(C(-2,1)\), \(C'(-1,-2)\); for \(D(-4,1)\), \(D'(-1,-4)\)

Step3: Apply translation

The translation vector is \(\langle3,4
angle\). The translation formula is \((x,y)\to(x + 3,y + 4)\)
For \(A'(-4,-4)\), \(A''(-4+3,-4 + 4)=A''(-1,0)\)
For \(B'(-4,-2)\), \(B''(-4+3,-2 + 4)=B''(-1,2)\)
For \(C'(-1,-2)\), \(C''(-1+3,-2 + 4)=C''(2,2)\)
For \(D'(-1,-4)\), \(D''(-1+3,-4 + 4)=D''(2,0)\)

Answer:

\(A'(-4,-4)\), \(B'(-4,-2)\), \(C'(-1,-2)\), \(D'(-1,-4)\)
\(A''(-1,0)\), \(B''(-1,2)\), \(C''(2,2)\), \(D''(2,0)\)