Question

the parallelogram ( tuvw ) is a dilation of the parallelogram ( tuvw ). 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 corresponding side lengths

Let's consider the vertical side of the parallelogram. For parallelogram \(TUVW\), take the vertical side from \(U(1,1)\) to \(V(1, - 1)\), the length is \(|1-(-1)| = 2\). For parallelogram \(T'U'V'W'\), take the vertical side from \(U'(5,5)\) to \(V'(5,-5)\), the length is \(|5 - (-5)|=10\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of corresponding side in pre - image}}\). So \(k=\frac{10}{2}=3\).

Another way:

Step1: Use coordinate transformation

If we consider a general point \((x,y)\) in the pre - image \(TUVW\) and its image \((x',y')\) in \(T'U'V'W'\). Let's take point \(U(1,1)\) and its image \(U'(5,5)\). The scale factor \(k\) for a dilation \((x,y)\to(kx,ky)\). We have \(kx = 5\) and \(x = 1\), \(ky=5\) and \(y = 1\). Solving \(k\times1=5\) (from the \(x\) - coordinate) or \(k\times1 = 5\) (from the \(y\) - coordinate), we get \(k = 3\).

Answer:

\(3\)