Question

1. triangle lmn with vertices l(-7, 4), m(-3, 0), and n(-8, 1): (a) translation: (x, y)→(x + 5, y - 6) (b) reflection: in the line y = -x l(, ) m(, ) n(, )

Explanation:

Step1: Perform translation on point L

For \(L(-7,4)\), using \((x,y)\to(x + 5,y - 6)\), we have \(x=-7,y = 4\). Then \(x'=-7 + 5=-2\) and \(y'=4-6=-2\). So \(L\) after translation is \(L_1(-2,-2)\).

Step2: Perform translation on point M

For \(M(-3,0)\), with \(x=-3,y = 0\), then \(x'=-3 + 5=2\) and \(y'=0-6=-6\). So \(M\) after translation is \(M_1(2,-6)\).

Step3: Perform translation on point N

For \(N(-8,1)\), with \(x=-8,y = 1\), then \(x'=-8 + 5=-3\) and \(y'=1-6=-5\). So \(N\) after translation is \(N_1(-3,-5)\).

Step4: Perform reflection on \(L_1\)

The rule for reflection in the line \(y=-x\) is \((x,y)\to(-y,-x)\). For \(L_1(-2,-2)\), \(x=-2,y=-2\), then \(x'=-(-2)=2\) and \(y'=-(-2)=2\). So \(L'\) is \((2,2)\).

Step5: Perform reflection on \(M_1\)

For \(M_1(2,-6)\), with \(x = 2,y=-6\), then \(x'=-(-6)=6\) and \(y'=-2\). So \(M'\) is \((6,-2)\).

Step6: Perform reflection on \(N_1\)

For \(N_1(-3,-5)\), with \(x=-3,y=-5\), then \(x'=-(-5)=5\) and \(y'=-(-3)=3\). So \(N'\) is \((5,3)\).

Answer:

\(L'(2,2)\)
\(M'(6,-2)\)
\(N'(5,3)\)