Question

1. rectangle defg with vertices d(-2, 7), e(2, 3), f(0, 1), and g(-4, 5):

a) translation along the rule (x, y)→(x + 6, y - 8)
b) reflection in the y - axis

1. triangle lmn with vertices l(6, 6), m(8, 8), and n(8, 3):

a) reflection in the line x = 5
b) 270° counterclockwise rotation about the origin

parallelogram pqrs with vertices p(-3, -4), q(0, -3), r(-1, -8), and s(-4, -9)
a) translation along the vector ⟨7, 0⟩
b) 90° counterclockwise rotation about (3, -2)

Explanation:

1. For rectangle \(DEFG\):

a) Translation

• Step1: Apply translation rule to point \(D\)

The translation rule is \((x,y)\to(x + 6,y - 8)\). For \(D(-2,7)\), we have \(x=-2\) and \(y = 7\). Then \(x'=-2+6 = 4\) and \(y'=7 - 8=-1\), so \(D'(4,-1)\).

• Step2: Apply translation rule to point \(E\)

For \(E(2,3)\), \(x = 2\) and \(y=3\). Then \(x'=2 + 6=8\) and \(y'=3-8=-5\), so \(E'(8,-5)\).

• Step3: Apply translation rule to point \(F\)

For \(F(0,1)\), \(x = 0\) and \(y = 1\). Then \(x'=0+6 = 6\) and \(y'=1 - 8=-7\), so \(F'(6,-7)\).

• Step4: Apply translation rule to point \(G\)

For \(G(-4,5)\), \(x=-4\) and \(y = 5\). Then \(x'=-4+6 = 2\) and \(y'=5 - 8=-3\), so \(G'(2,-3)\).

b) Reflection in the \(y\) - axis

The rule for reflection in the \(y\) - axis is \((x,y)\to(-x,y)\).

• For \(D'(4,-1)\), the new point \(D''(-4,-1)\).
• For \(E'(8,-5)\), the new point \(E''(-8,-5)\).
• For \(F'(6,-7)\), the new point \(F''(-6,-7)\).
• For \(G'(2,-3)\), the new point \(G''(-2,-3)\).

2. For triangle \(LMN\):

a) Reflection in the line \(x = 5\)

The distance between a point \((x,y)\) and the line \(x = 5\) is \(d=\vert x - 5\vert\). The new \(x\) - coordinate after reflection in the line \(x = 5\) is \(x'=5+(5 - x)=10 - x\) and \(y'\) remains the same.

• For \(L(6,6)\), \(x'=10 - 6 = 4\) and \(y'=6\), so \(L'(4,6)\).
• For \(M(8,8)\), \(x'=10 - 8 = 2\) and \(y'=8\), so \(M'(2,8)\).
• For \(N(8,3)\), \(x'=10 - 8 = 2\) and \(y'=3\), so \(N'(2,3)\).

b) \(270^{\circ}\) counter - clockwise rotation about the origin

The rule for a \(270^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(y,-x)\).

• For \(L'(4,6)\), \(L''(6,-4)\).
• For \(M'(2,8)\), \(M''(8,-2)\).
• For \(N'(2,3)\), \(N''(3,-2)\).

3. For parallelogram \(PQRS\):

a) Translation along the vector \(\langle7,0

angle\)
The rule for translation along the vector \(\langle7,0
angle\) is \((x,y)\to(x + 7,y)\).

• For \(P(-3,-4)\), \(x'=-3+7 = 4\) and \(y'=-4\), so \(P'(4,-4)\).
• For \(Q(0,-3)\), \(x'=0 + 7=7\) and \(y'=-3\), so \(Q'(7,-3)\).
• For \(R(-1,-8)\), \(x'=-1+7 = 6\) and \(y'=-8\), so \(R'(6,-8)\).
• For \(S(-4,-9)\), \(x'=-4+7 = 3\) and \(y'=-9\), so \(S'(3,-9)\).

b) \(90^{\circ}\) counter - clockwise rotation about \((3,-2)\)

First, translate the point \((x,y)\) so that the center of rotation \((3,-2)\) is at the origin: \((x_1,y_1)=(x - 3,y + 2)\). Then apply the \(90^{\circ}\) counter - clockwise rotation rule \((x_1,y_1)\to(-y_1,x_1)\). Then translate back: \((x_2,y_2)=(-y_1+3,x_1 - 2)\).

• For \(P'(4,-4)\):
• \((x_1,y_1)=(4 - 3,-4 + 2)=(1,-2)\).
• After rotation \((x_3,y_3)=(2,1)\).
• After translation back \(P''(2 + 3,1-2)=(5,-1)\).
• For \(Q'(7,-3)\):
• \((x_1,y_1)=(7 - 3,-3 + 2)=(4,-1)\).
• After rotation \((x_3,y_3)=(1,4)\).
• After translation back \(Q''(1 + 3,4-2)=(4,2)\).
• For \(R'(6,-8)\):
• \((x_1,y_1)=(6 - 3,-8 + 2)=(3,-6)\).
• After rotation \((x_3,y_3)=(6,3)\).
• After translation back \(R''(6 + 3,3-2)=(9,1)\).
• For \(S'(3,-9)\):
• \((x_1,y_1)=(3 - 3,-9 + 2)=(0,-7)\).
• After rotation \((x_3,y_3)=(7,0)\).
• After translation back \(S''(7 + 3,0-2)=(10,-2)\).

Answer:

1. Rectangle \(DEFG\):

• After translation: \(D'(4,-1)\), \(E'(8,-5)\), \(F'(6,-7)\), \(G'(2,-3)\)
• After reflection in \(y\) - axis: \(D''(-4,-1)\), \(E''(-8,-5)\), \(F''(-6,-7)\), \(G''(-2,-3)\)

1. Triangle \(LMN\):

• After reflection in \(x = 5\): \(L'(4,6)\), \(M'(2,8)\), \(N'(2,3)\)
• After \(270^{\circ}\) rotation about origin: \(L''(6,-4)\), \(M''(8,-2)\), \(N''(3,-2)\)

1. Parallelogram \(PQRS\):

• After translation: \(P'(4,-4)\), \(Q'(7,-3)\), \(R'(6,-8)\), \(S'(3,-9)\)
• After \(90^{\circ}\) rotation about \((3,-2)\): \(P''(5,-1)\), \(Q''(4,2)\), \(R''(9,1)\), \(S''(10,-2)\)