Question

graph \\( \triangle l m n \\) with vertices \\( l ( 1,6 ) , m ( - 2,4 ) \\), and \\( n ( 3,2 ) \\) and its image after the composition.
translation: \\( ( x , y ) \
ightarrow ( x + 1 , y + 4 ) \\)
rotation: \\( 180 ^ { \circ } \\) about the origin
polgon undo redo × reset

Explanation:

Step1: Find coordinates after translation

For point \(L(1,6)\):
Using the translation \((x,y)\to(x + 1,y + 4)\), we have \(x=1+1 = 2\), \(y=6 + 4=10\). So \(L'(2,10)\)
For point \(M(-2,4)\):
\(x=-2+1=-1\), \(y=4 + 4 = 8\). So \(M'(-1,8)\)
For point \(N(3,2)\):
\(x=3+1=4\), \(y=2+4 = 6\). So \(N'(4,6)\)

Step2: Find coordinates after rotation

The rule for a \(180^{\circ}\) rotation about the origin is \((x,y)\to(-x,-y)\)
For \(L'(2,10)\):
\(x=-2\), \(y=-10\). So \(L''(-2,-10)\)
For \(M'(-1,8)\):
\(x = 1\), \(y=-8\). So \(M''(1,-8)\)
For \(N'(4,6)\):
\(x=-4\), \(y=-6\). So \(N''(-4,-6)\)

Answer:

The coordinates of the image of \(\triangle LMN\) after the composition (translation then \(180^{\circ}\) rotation about the origin) are \(L''(-2,-10)\), \(M''(1,-8)\) and \(N''(-4,-6)\)