Question

the vertices of triangle pqr are p(2,4), q(6,0), and r(7,2). find the coordinates of the image after the following composition:
translation: (x, y) → (x - 3, y + 2)
rotation: 90° counterclockwise about the origin
p(
q(
r(

Explanation:

Step1: Apply the translation

For point \(P(2,4)\):
\(x = 2-3=-1\), \(y = 4 + 2=6\). So \(P_1(-1,6)\)
For point \(Q(6,0)\):
\(x = 6-3 = 3\), \(y=0 + 2=2\). So \(Q_1(3,2)\)
For point \(R(7,2)\):
\(x=7 - 3=4\), \(y=2+2 = 4\). So \(R_1(4,4)\)

Step2: Apply the rotation

The rule for a \(90^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-y,x)\)
For \(P_1(-1,6)\):
\(x=-6\), \(y=-1\). So \(P'(-6,-1)\)
For \(Q_1(3,2)\):
\(x=-2\), \(y = 3\). So \(Q'(-2,3)\)
For \(R_1(4,4)\):
\(x=-4\), \(y = 4\). So \(R'(-4,4)\)

Answer:

\(P'(-6,-1)\), \(Q'(-2,3)\), \(R'(-4,4)\)