Question

1. graph \\( \triangle def \\) with \\( d(4,4) \\), \\( e(6,2) \\) and \\( f(2,0) \\). graph the image of \\( \triangle def \\) under \\( r_{x - axis} \circ r_{180} \\).

Explanation:

Step1: Find the coordinates after \( R_{180} \)

The rule for a \( 180^{\circ} \) rotation about the origin \( (x,y)\to(-x,-y) \).
For point \( D(4,4) \), after \( R_{180} \), \( D'(- 4,-4) \).
For point \( E(6,2) \), after \( R_{180} \), \( E'(-6,-2) \).
For point \( F(2,0) \), after \( R_{180} \), \( F'(-2,0) \).

Step2: Find the coordinates after \( r_{x - axis} \)

The rule for a reflection over the \( x - axis \) is \( (x,y)\to(x,-y) \).
For point \( D'(-4,-4) \), after \( r_{x - axis} \), \( D''(-4,4) \).
For point \( E'(-6,-2) \), after \( r_{x - axis} \), \( E''(-6,2) \).
For point \( F'(-2,0) \), after \( r_{x - axis} \), \( F''(-2,0) \).

Answer:

Plot the original triangle \( \triangle DEF \) with \( D(4,4) \), \( E(6,2) \), \( F(2,0) \) and the transformed triangle \( \triangle D''E''F'' \) with \( D''(-4,4) \), \( E''(-6,2) \), \( F''(-2,0) \) on the coordinate - plane.