Question

in the similarity transformation of \\( \triangle acb \\) to \\( \triangle dfe, \triangle acb \\) was dilated by a scale factor of ?, reflected across the , and moved through the translation .

Explanation:

Step1: Determine the scale factor

To find the scale factor, we compare the lengths of corresponding sides. Let's assume a side in \(\triangle ACB\) and the corresponding side in \(\triangle DFE\).
Suppose we consider the vertical side. In \(\triangle ACB\), if we assume a vertical - like side (for example, the distance from \(A(0,0)\) to \(B(-6,3)\) in terms of vertical component, but a better way is to use the distance formula. Let's use the horizontal - like side. The length of \(AC\): \(A(0,0)\) and \(C(- 3,0)\), so \(AC = 3\). The length of \(DF\): \(D(2,-1)\) and \(F(0,-1)\), so \(DF = 2\). But wait, another approach: if we use the ratio of the lengths of \(AB\) and \(DE\).
The coordinates of \(A(0,0)\), \(B(-6,3)\), \(D(2,-1)\), \(E(0,-2)\).
The length of \(AB=\sqrt{(-6 - 0)^2+(3 - 0)^2}=\sqrt{36 + 9}=\sqrt{45}=3\sqrt{5}\).
The length of \(DE=\sqrt{(0 - 2)^2+(-2+1)^2}=\sqrt{4 + 1}=\sqrt{5}\).
The scale factor \(k=\frac{DE}{AB}=\frac{\sqrt{5}}{3\sqrt{5}}=\frac{1}{3}\).

Step2: Determine the reflection axis

By observing the orientation of the triangles. \(\triangle ACB\) is above the \(x\) - axis and \(\triangle DFE\) is below the \(x\) - axis. When a figure is reflected across the \(x\) - axis, the \(x\) - coordinate of each point remains the same and the \(y\) - coordinate is multiplied by \(-1\).

Step3: Determine the translation

After dilation and reflection, assume a general point \((x,y)\) after dilation with scale factor \(\frac{1}{3}\) and reflection across the \(x\) - axis becomes \((\frac{1}{3}x,-\frac{1}{3}y)\).
For point \(A(0,0)\): after dilation and reflection it is \((0,0)\), and then to get to the position in \(\triangle DFE\) (for example, if we consider the centroid or a point transformation).
Let's take point \(B(-6,3)\). After dilation with scale factor \(\frac{1}{3}\): \((- 2,1)\), after reflection across the \(x\) - axis: \((-2,-1)\). To get to \(D(2,-1)\), we use the translation rule \((x,y)\to(x + 4,y)\)

Answer:

The scale factor is \(\frac{1}{3}\), reflected across the \(x\) - axis, and the translation is \((x,y)\to(x + 4,y)\)