Question

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

Explanation:

Step1: Calculate the scale factor

The length of \(AB\) is \(6\) units (\(B\) is at \(x = 2\) and \(A\) is at \(x=-4\), \(|2 - (-4)|=6\)). The length of \(DE\) is \(2\) units (\(D\) is at \(x = 3\) and \(E\) is at \(x = 5\), \(|5 - 3| = 2\)). The scale factor \(k=\frac{DE}{AB}=\frac{2}{6}=\frac{1}{3}\).

Step2: Determine the reflection axis

The \(y\) - coordinate of points in \(\triangle ABC\) is negative (below \(x\) - axis) and in \(\triangle EDF\) is non - negative (above or on \(x\) - axis). So, it is reflected across the \(x\) - axis.

Step3: Find the translation

Let's consider a point. For example, point \(B(2,-2)\). After reflection across the \(x\) - axis, it becomes \((2,2)\). Then, to get to a point in \(\triangle EDF\) (say \(E(5,1)\) is not the right example. Let's use the mid - point approach. The mid - point of \(AB\) is \((\frac{-4 + 2}{2},\frac{-2-4}{2})=(-1,-3)\). The mid - point of \(DE\) is \((\frac{3 + 5}{2},\frac{1+1}{2})=(4,1)\). The translation rule \((x,y)\to(x + 5,y+4)\) (from \(x=-1\) to \(x = 4\) is \(x+5\), from \(y=-3\) to \(y = 1\) is \(y + 4\))

Answer:

The scale factor is \(\frac{1}{3}\), reflected across the \(x\) - axis, and the translation is \((x,y)\to(x + 5,y+4)\)