Question

directions: graph and label each figure and its image under the given dilation.

1. parallelogram jklm with vertices j(0,4), k(3,5), l(1,0), and m(-2,-1); k = 2
2. square pqrs with vertices p(1,-2), q(2,-1), r(3,-2), and s(2,-3); k = 5
3. triangle wxy with vertices w(-8,16), x(8,12), and y(-4,8); k = 1/4
4. trapezoid abcd with vertices a(-18,0), b(-15,9), c(-6,12), and d(0,6); k = 1/3
5. rhombus stuv with vertices s(1,1), t(4,2), u(3,-1), and v(0,-2); k = 4
6. triangle fgh with vertices f(-4,0), g(-2,-6), and h(-6,-4); k = 5/2

Explanation:

Step1: Dilation formula

When a point \((x,y)\) is dilated with a scale factor \(k\) about the origin, the new point \((x',y')\) is given by \((x',y')=(kx,ky)\)

Step2: For square \(PQRS\) with \(k = 5\)

• For \(P(1,-2)\): \(P'=(5\times1,5\times(-2))=(5,-10)\)
• For \(Q(2,-1)\): \(Q'=(5\times2,5\times(-1))=(10,-5)\)
• For \(R(3,-2)\): \(R'=(5\times3,5\times(-2))=(15,-10)\)
• For \(S(2,-3)\): \(S'=(5\times2,5\times(-3))=(10,-15)\)

Step3: For trapezoid \(ABCD\) with \(k=\frac{1}{3}\)

• For \(A(-18,0)\): \(A'=(\frac{1}{3}\times(-18),\frac{1}{3}\times0)=(-6,0)\)
• For \(B(-15,9)\): \(B'=(\frac{1}{3}\times(-15),\frac{1}{3}\times9)=(-5,3)\)
• For \(C(-6,12)\): \(C'=(\frac{1}{3}\times(-6),\frac{1}{3}\times12)=(-2,4)\)
• For \(D(0,6)\): \(D'=(\frac{1}{3}\times0,\frac{1}{3}\times6)=(0,2)\)

Step4: For triangle \(FGH\) with \(k = \frac{5}{2}\)

• For \(F(-4,0)\): \(F'=(\frac{5}{2}\times(-4),\frac{5}{2}\times0)=(-10,0)\)
• For \(G(-2,-6)\): \(G'=(\frac{5}{2}\times(-2),\frac{5}{2}\times(-6))=(-5,-15)\)
• For \(H(-6,-4)\): \(H'=(\frac{5}{2}\times(-6),\frac{5}{2}\times(-4))=(-15,-10)\)

Answer:

1. For parallelogram \(JKLM\) with \(k = 2\):

\(J'(0\times2,4\times2)=(0,8)\), \(K'(3\times2,5\times2)=(6,10)\), \(L'(1\times2,0\times2)=(2,0)\), \(M'(-2\times2,-1\times2)=(-4,-2)\)

1. Square \(PQRS\): \(P'(5,-10)\), \(Q'(10,-5)\), \(R'(15,-10)\), \(S'(10,-15)\)
2. Triangle \(WXY\) with \(k=\frac{1}{4}\):

\(W'(-8\times\frac{1}{4},16\times\frac{1}{4})=(-2,4)\), \(X'(8\times\frac{1}{4},12\times\frac{1}{4})=(2,3)\), \(Y'(-4\times\frac{1}{4},8\times\frac{1}{4})=(-1,2)\)

1. Trapezoid \(ABCD\): \(A'(-6,0)\), \(B'(-5,3)\), \(C'(-2,4)\), \(D'(0,2)\)
2. Rhombus \(STUV\) with \(k = 4\):

\(S'(1\times4,1\times4)=(4,4)\), \(T'(4\times4,2\times4)=(16,8)\), \(U'(3\times4,-1\times4)=(12,-4)\), \(V'(0\times4,-2\times4)=(0,-8)\)

1. Triangle \(FGH\): \(F'(-10,0)\), \(G'(-5,-15)\), \(H'(-15,-10)\)