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 side in the original triangle and the corresponding side in the dilated triangle

Let's consider the side \(QR\). The coordinates of \(Q(-1,0)\) and \(R(0, - 2)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \(Q(-1,0)\) and \(R(0,-2)\), we have \(QR=\sqrt{(0 + 1)^2+(-2-0)^2}=\sqrt{1 + 4}=\sqrt{5}\).
The coordinates of \(Q'(-10,0)\) and \(R'(0,-10)\). Using the distance formula, for \(Q'(-10,0)\) and \(R'(0,-10)\), we have \(Q'R'=\sqrt{(0 + 10)^2+(-10 - 0)^2}=\sqrt{100+100}=\sqrt{200}=10\sqrt{2}\).
Another way (since it's a dilation from the origin):
Take a vertical or horizontal segment. For example, the vertical segment from \(R(0,-2)\) to \(Q(-1,0)\) (using the ratio of coordinates). If we consider the \(y -\)coordinate of \(R\) is \(y=-2\) and \(y'=-10\) for \(R'\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the ratio of the length of a side of the dilated figure to the length of the corresponding side of the original figure.
If we use the \(y -\)coordinates (since dilation is centered at the origin, we can use the ratio of corresponding coordinates). Let \((x,y)\) be a point of the original triangle and \((x',y')\) be the point of the dilated triangle. The formula for dilation centered at the origin is \((x',y')=(kx,ky)\).
Take the point \(R(0,-2)\) and \(R'(0,-10)\). We know that \(y'=ky\), so \(k=\frac{y'}{y}\). Substituting \(y=-2\) and \(y'=-10\), we get \(k = 5\).

Answer:

\(5\)