Question

1. rhombus pqrs with vertices ( p(-1,-3), q(3,-4), r(4,-8) ), and ( s(0,-7) ):

a) translation along the vector ( (4,10) )
b) ( 90^{circ} ) counterclockwise rotation about the origin

1. parallelogram cdef with vertices ( c(2,-1), d(7,1), e(6,-2) ), and ( f(1,-4) ):

a) reflection in the line ( y = x )
b) translated along the rule ( (x,y)\to(x - 3,y - 7) )

1. triangle lmn with vertices ( l(0,3), m(3,4) ), and ( n(1,2) ):

a) ( 180^{circ} ) rotation about the origin
b) dilation with scale factor of 2 using the origin as the center

Explanation:

Problem 3a

Step1: Apply translation rule

For a point \((x,y)\) translated along the vector \((a,b)\), the new point is \((x + a,y + b)\). Here \(a = 4\) and \(b=10\).
For \(P(-1,-3)\): \(x=-1,y = - 3\), \(P'=(-1+4,-3 + 10)=(3,7)\)

Step2: Apply translation rule to \(Q\)

For \(Q(3,-4)\): \(x = 3,y=-4\), \(Q'=(3 + 4,-4+10)=(7,6)\)

Step3: Apply translation rule to \(R\)

For \(R(4,-8)\): \(x = 4,y=-8\), \(R'=(4 + 4,-8+10)=(8,2)\)

Step4: Apply translation rule to \(S\)

For \(S(0,-7)\): \(x = 0,y=-7\), \(S'=(0 + 4,-7+10)=(4,3)\)

Step1: Apply \(90^{\circ}\) counter - clockwise rotation rule

The rule for a \(90^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-y,x)\)
For \(P(-1,-3)\): \(x=-1,y = - 3\), \(P'=(3,-1)\)

Step2: Apply rotation rule to \(Q\)

For \(Q(3,-4)\): \(x = 3,y=-4\), \(Q'=(4,3)\)

Step3: Apply rotation rule to \(R\)

For \(R(4,-8)\): \(x = 4,y=-8\), \(R'=(8,4)\)

Step4: Apply rotation rule to \(S\)

For \(S(0,-7)\): \(x = 0,y=-7\), \(S'=(7,0)\)

Step1: Apply reflection in \(y = x\) rule

The rule for reflection in the line \(y=x\) is \((x,y)\to(y,x)\)
For \(C(2,-1)\): \(x = 2,y=-1\), \(C'=(-1,2)\)

Step2: Apply reflection rule to \(D\)

For \(D(7,1)\): \(x = 7,y = 1\), \(D'=(1,7)\)

Step3: Apply reflection rule to \(E\)

For \(E(6,-2)\): \(x = 6,y=-2\), \(E'=(-2,6)\)

Step4: Apply reflection rule to \(F\)

For \(F(1,-4)\): \(x = 1,y=-4\), \(F'=(-4,1)\)

Answer:

\(P'(3,7)\), \(Q'(7,6)\), \(R'(8,2)\), \(S'(4,3)\)

Problem 3b