Question

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

Explanation:

Step1: Calculate the scale factor

First, find the length of a side in \(\triangle ABC\) and the corresponding side in \(\triangle DEF\).
Let's take \(AB\) in \(\triangle ABC\). The coordinates of \(A=(3,1)\) and \(B=(1,1)\), so \(AB = 3 - 1=2\).
The coordinates of \(D=(2,-1)\) and \(E=(-4,-1)\), so \(DE=2-(-4) = 6\).
The scale factor \(k=\frac{DE}{AB}\). Since \(DE = 6\) and \(AB = 2\), \(k = 3\).

Step2: Determine the reflection axis

Looking at the \(y -\)coordinates of corresponding points. For example, point \(A=(3,1)\) and \(D=(2,-1)\), point \(B=(1,1)\) and \(E=(-4,-1)\), point \(C=(3,2)\) and \(F=(2,-3)\).
The mid - line between \(y = 1\) (for points in \(\triangle ABC\)) and \(y=-1\) (for some points in \(\triangle DEF\)) is \(y = 0\) (the \(x -\)axis). So \(\triangle ABC\) is reflected across the \(x -\)axis.

Step3: Determine the translation

After dilation and reflection, assume the translation is \((x,y)\to(x + a,y + b)\).
Take a point, say \(A=(3,1)\). After dilation by scale factor \(3\) and reflection across the \(x -\)axis, the point becomes \((9,-1)\). But \(D=(2,-1)\).
We have the equation \(9 + a=2\) and \(-1 + b=-1\). Solving \(a=2 - 9=-7\) and \(b=-1+1 = 0\). The translation is \((x,y)\to(x - 7,y)\)

Answer:

The scale factor is \(3\), reflected across the \(x -\)axis, and the translation is \((x,y)\to(x - 7,y)\)