Question

the trapezoid ( stuv ) is a dilation of the trapezoid ( stuv ). 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 corresponding side in both trapezoids

Let's consider the vertical side \(TU\) and \(T'U'\).
The coordinates of \(T\) are \((-5,5)\) and \(U\) are \((5,0)\). The length of \(TU\) can be found using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). But since it's a vertical - like side (for simplicity, we can count the units).
The length of \(TU\): The \(y\) - coordinate of \(T\) is \(5\) and of \(U\) is \(0\), and the \(x\) - coordinate of \(T\) is \(- 5\) and of \(U\) is \(5\). The length of \(TU\) (using the distance formula \(d=\sqrt{(5+5)^2+(0 - 5)^2}=\sqrt{100 + 25}=\sqrt{125}=5\sqrt{5}\), but we can also count the units from the graph. The vertical distance from \(T\) to \(U\) (by counting the grid units): \(T\) is at \(y = 5\) and \(U\) is at \(y = 0\) (and considering the horizontal alignment, the length of \(TU\) (counting the number of units) is \(5\) units (from \(y = 5\) to \(y = 0\) for the non - horizontal part).
The coordinates of \(T'\) are \((-7,7)\) and \(U'\) are \((7,0)\). The length of \(T'U'\) (counting the units from the graph): \(T'\) is at \(y = 7\) and \(U'\) is at \(y = 0\) (and considering the horizontal alignment), the length of \(T'U'\) (counting the number of units) is \(7\) units (from \(y = 7\) to \(y = 0\) for the non - horizontal part).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of a side in the image}}{\text{length of the corresponding side in the pre - image}}\).
Let the pre - image be trapezoid \(STUV\) and the image be trapezoid \(S'T'U'V'\).
If we take the side \(TU\) (pre - image) and \(T'U'\) (image). The length of \(TU = 5\) (counting units) and the length of \(T'U'=7\) (counting units). But wait, we made a mistake above. Let's use another pair of corresponding sides.
Take the side \(SU\). \(S\) is at \((-5,-5)\) and \(U\) is at \((5,0)\). The length of \(SU\) (counting units): The vertical distance from \(y=-5\) to \(y = 0\) is \(5\) units and the horizontal distance from \(x=-5\) to \(x = 5\) is \(10\) units. Using the Pythagorean theorem \(d=\sqrt{(5 + 5)^2+(0+5)^2}=\sqrt{100 + 25}=\sqrt{125}\). But using the ratio of corresponding sides.
Let's use the horizontal - like sides (since dilation is uniform).
Take the side \(UV\). \(U=(5,0)\) and \(V=(5,-5)\), length \(UV = 5\) units (vertical side). \(U'=(7,0)\) and \(V'=(7,-7)\), length \(U'V'=7\) units (vertical side).
The scale factor \(k=\frac{\text{length of }U'V'}{\text{length of }UV}\)
\(k=\frac{7}{5}\)

Answer:

\(\frac{7}{5}\)