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 the length of corresponding sides

Let's consider the horizontal distance from \(T\) to \(U\) and from \(T'\) to \(U'\).
For parallelogram \(TUVW\), if \(T(- 2,0)\) and \(U(1,1)\), the horizontal distance (change in \(x\)-coordinate) from \(T\) to \(U\) is \(1-(-2)=3\).
For parallelogram \(T'U'V'W'\), if \(T'(-10,0)\) and \(U'(5,5)\), the horizontal distance (change in \(x\)-coordinate) from \(T'\) to \(U'\) is \(5 - (-10)=15\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of side in pre - image}}\).
Using the horizontal side lengths, \(k = \frac{15}{3}=5\).

Another way:
Take a vertical side. For example, consider the vertical distance from \(U\) to \(V\) and from \(U'\) to \(V'\).
For parallelogram \(TUVW\), if \(U(1,1)\) and \(V(1,-1)\), the vertical distance (change in \(y\)-coordinate) is \(1-(-1) = 2\).
For parallelogram \(T'U'V'W'\), if \(U'(5,5)\) and \(V'(5,-5)\), the vertical distance (change in \(y\)-coordinate) is \(5-(-5)=10\).
Using the vertical side lengths, \(k=\frac{10}{2}=5\).

Answer:

\(5\)