Question

aph \\( \triangle l m n \\) with vertices \\( l ( 1,6 ) , m ( - 2,4 ) \\), and \\( n ( 3,2 ) \\) and its image after the composition
anslation: \\( ( x , y ) \
ightarrow ( x + 1 , y + 4 ) \\)
otation: \\( 180 ^ { \circ } \\) about the origin

Explanation:

Step1: Apply translation

For point \(L(1,6)\):
\(x = 1+1=2\), \(y=6 + 4=10\), 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: Apply \(180^{\circ}\) rotation about the origin

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

Answer:

The vertices of the image triangle are \(L(-2,-10)\), \(M(1,-8)\), \(N(-4,-6)\)