Question

in the similarity transformation of \\( \triangle abc \\) to \\( \triangle dfe \\), \\( \triangle abc \\) was dilated by a scale factor of \\( 1/2 \\), reflected across the ?, and moved through the translation .

Explanation:

Step1: Analyze the reflection

Looking at the positions of the triangles, the \(y -\)coordinates of corresponding points change sign. For example, if we consider a general point \((x,y)\) in \(\triangle ABC\) and its corresponding point in \(\triangle DFE\) after reflection. A reflection across the \(x -\)axis changes the sign of the \(y -\)coordinate of a point. The transformation rule for a reflection across the \(x -\)axis is \((x,y)\to(x, - y)\).

Step2: Analyze the translation

Let's assume a point \(A(x_1,y_1)\) in \(\triangle ABC\). After dilation by a scale factor of \(\frac{1}{2}\) and reflection across the \(x -\)axis, we then need to find the translation.
If we consider the center - like movement. Suppose we take a vertex of \(\triangle ABC\). Let's assume after dilation and reflection, we observe the horizontal and vertical shifts.
For a point \((x,y)\) after dilation (\((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\)) and reflection across the \(x -\)axis \((\frac{1}{2}x,-\frac{1}{2}y)\), and then translation.
If we consider the movement from the lower - left triangle (\(\triangle ABC\)) to the upper - right triangle (\(\triangle DFE\)), we can use the formula for translation \((x,y)\to(x + h,y + k)\).
By comparing the positions of non - origin related vertices (to avoid confusion with dilation center assumptions), we find that the translation rule is \((x,y)\to(x+4,y + 2)\)

Answer:

The first blank: \(x -\)axis. The translation: \((x,y)\to(x + 4,y+2)\)