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:
reflection: in the x - axis
rotation: 270° counterclockwise about the origin
p(
q(
r(

Explanation:

Step1: Reflection in the x - axis

When a point \((x,y)\) is reflected in the \(x\) - axis, the transformation rule is \((x,y)\to(x, - y)\).
For \(P(2,4)\), after reflection in the \(x\) - axis, \(P_1(2,-4)\).
For \(Q(6,0)\), after reflection in the \(x\) - axis, \(Q_1(6,0)\).
For \(R(7,2)\), after reflection in the \(x\) - axis, \(R_1(7,-2)\).

Step2: Rotation \(270^{\circ}\) counter - clockwise about the origin

The rule for a \(270^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(y,-x)\).
For \(P_1(2,-4)\), after rotation: \(x = 2,y=-4\), then \(P''(-4,- 2)\) (since \(x_1=y=-4\) and \(y_1=-x=-2\)).
For \(Q_1(6,0)\), after rotation: \(x = 6,y = 0\), then \(Q''(0,-6)\) (since \(x_1=y = 0\) and \(y_1=-x=-6\)).
For \(R_1(7,-2)\), after rotation: \(x = 7,y=-2\), then \(R''(-2,-7)\) (since \(x_1=y=-2\) and \(y_1=-x=-7\)).

Answer:

\(P''(-4,-2)\), \(Q''(0,-6)\), \(R''(-2,-7)\)