Question

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

Explanation:

Step1: Calculate the scale factor

The length of \(BF\) (vertical side of \(\triangle DFE\)) is \(1\) unit, and the length of \(BC\) (vertical side of \(\triangle ABC\)) is \(3\) units. The scale factor \(k\) is given by the ratio of the corresponding side lengths of the image to the pre - image. So, \(k=\frac{1}{3}\)

Step2: Determine the reflection axis

By observing the positions of the triangles, \(\triangle ABC\) is reflected across the \(x -\)axis to get a triangle in the upper - half plane (before dilation and translation).

Step3: Determine the translation

Let's assume a general point \((x,y)\) after reflection across the \(x -\)axis is \((x, - y)\). After dilation with scale factor \(\frac{1}{3}\), a point \((x,y)\) of \(\triangle ABC\) becomes \((\frac{1}{3}x,-\frac{1}{3}y)\).
If we consider a point \(B(0, - 2)\) in \(\triangle ABC\):

• After reflection across the \(x -\)axis: \((0,2)\)
• After dilation with scale factor \(\frac{1}{3}\): \((0,\frac{2}{3})\)
• To get to \(F(2,1)\) (assuming we consider the translation rule \((x,y)\to(x + h,y + k)\)), we can also use another approach.

The center of dilation is the origin.
Let’s take \(A(-4,-2)\):

• After reflection across \(x -\)axis: \((-4,2)\)
• After dilation with scale factor \(\frac{1}{3}\): \((-\frac{4}{3},\frac{2}{3})\)
• To get to \(D(1,1)\), we use the translation rule \((x,y)\to(x + \frac{7}{3},y+\frac{1}{3})\)

Answer:

Scale factor: \(\frac{1}{3}\), reflected across the \(x -\)axis.