Question

write the coordinates of the vertices after a dilation with a scale factor of 3, centered at the origin.

Explanation:

1. First, identify the original - vertex coordinates:

• From the graph, assume the vertices of the rectangle are \(R(-1,-1)\), \(S(-1,3)\), \(T(2,3)\), and \(U(2,-1)\).
• The formula for dilation centered at the origin \((0,0)\) with a scale - factor \(k\) is \((x,y)\to(kx,ky)\). Here, \(k = 3\).

1. Calculate the new coordinates for each vertex:

• For vertex \(R(-1,-1)\):
• Multiply the \(x\) and \(y\) coordinates by the scale - factor \(3\).
• \(x_{new}=3\times(-1)=-3\), \(y_{new}=3\times(-1)=-3\). So the new coordinates of \(R\) are \(R'(-3,-3)\).
• For vertex \(S(-1,3)\):
• \(x_{new}=3\times(-1)=-3\), \(y_{new}=3\times3 = 9\). So the new coordinates of \(S\) are \(S'(-3,9)\).
• For vertex \(T(2,3)\):
• \(x_{new}=3\times2 = 6\), \(y_{new}=3\times3 = 9\). So the new coordinates of \(T\) are \(T'(6,9)\).
• For vertex \(U(2,-1)\):
• \(x_{new}=3\times2 = 6\), \(y_{new}=3\times(-1)=-3\). So the new coordinates of \(U\) are \(U'(6,-3)\).

Step1: Identify original coordinates

Assume \(R(-1,-1)\), \(S(-1,3)\), \(T(2,3)\), \(U(2,-1)\)

Step2: Apply dilation formula for \(R\)

\((-1,-1)\to(3\times(-1),3\times(-1))=(-3,-3)\)

Step3: Apply dilation formula for \(S\)

\((-1,3)\to(3\times(-1),3\times3)=(-3,9)\)

Step4: Apply dilation formula for \(T\)

\((2,3)\to(3\times2,3\times3)=(6,9)\)

Step5: Apply dilation formula for \(U\)

\((2,-1)\to(3\times2,3\times(-1))=(6,-3)\)

Answer:

\(R'(-3,-3)\), \(S'(-3,9)\), \(T'(6,9)\), \(U'(6,-3)\)