Question

the rectangle ( rstu ) is a dilation of the rectangle ( rstu ). what is the scale factor of the dilation?
simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

Explanation:

Step1: Find the length of a side of the original rectangle

The length of \(ST\) in rectangle \(RSTU\): \(S(3,1)\), \(T(3, - 1)\). Using the distance formula (for vertical points \(x\) - coordinate is same) \(d=\vert y_2 - y_1\vert\). So \(ST=\vert1-(-1)\vert = 2\).

Step2: Find the length of the corresponding side of the dilated rectangle

The length of \(S'T'\) in rectangle \(R'S'T'U'\): \(S'(9,3)\), \(T'(9, - 3)\). Using the distance formula (for vertical points \(x\) - coordinate is same) \(d=\vert y_2 - y_1\vert\). So \(S'T'=\vert3-(-3)\vert = 6\).

Step3: Calculate the scale factor

The scale factor \(k\) of a dilation is given by \(k=\frac{\text{length of side of dilated figure}}{\text{length of side of original figure}}\). So \(k = \frac{S'T'}{ST}=\frac{6}{2}=3\).

Answer:

\(3\)