Question

the triangle $qrs$ is a dilation of the triangle $qrs$. 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 triangles

Let's use the vertical side \(QQ'\) and \(Q'Q''\). The length of \(QR\) (from \(Q(-1,0)\) to \(R(0, - 2)\)): Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \(Q(-1,0)\) and \(R(0,-2)\), \(d_{QR}=\sqrt{(0 + 1)^2+(-2-0)^2}=\sqrt{1 + 4}=\sqrt{5}\). Another way (since it's a vertical - like side in a grid): Count the units. The length of \(QR\): from \(y = 0\) to \(y=-2\) (for the vertical part of the side related to the dilation). The length of \(Q'R'\): from \(y = 0\) ( \(Q'(-10,0)\)) to \(y=-10\) ( \(R'(0,-10)\)).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by \(k=\frac{\text{length of side in image}}{\text{length of corresponding side in pre - image}}\).
If we consider the vertical segments (since dilation is uniform). The length of the segment from \(Q\) to \(R\) (vertical component): The \(y\) - coordinate of \(Q\) is \(0\) and of \(R\) is \(-2\), so the length (distance) related to dilation is \(|0-(-2)| = 2\). The length of the segment from \(Q'\) to \(R'\): The \(y\) - coordinate of \(Q'\) is \(0\) and of \(R'\) is \(-10\), so the length (distance) is \(|0-(-10)|=10\).
\(k = 5\)

Another approach:
If we use the formula for dilation. If a point \(P(x,y)\) is dilated to \(P'(x',y')\) with a scale factor \(k\) from the origin \((0,0)\) (assuming the center of dilation is the origin, since when \(x = 0,y = 0\) for \(R\) and \(R'\), \(x=-1,y = 0\) for \(Q\) and \(x=-10,y = 0\) for \(Q'\)).
For a point \(Q(-1,0)\) dilated to \(Q'(-10,0)\). Using the formula \(x'=kx\) and \(y'=ky\). Substituting \(x=-1,x'=-10,y = 0,y'=0\) into \(x'=kx\), we get \(-10=k\times(-1)\)

Answer:

\(5\)