Question

choose a similarity transformation that maps the blue figure to the red figure.
one possible similarity transformation is a reflection in the
axis followed by a dilation centered at the origin with a scale factor of

Explanation:

Step1: Analyze the reflection

First, reflect the blue triangle over the \(x -\)axis. A reflection over the \(x -\)axis changes the \(y -\)coordinate of each point \((x,y)\) to \((x, - y)\).

Step2: Analyze the dilation

After reflection, consider the dilation. Let's assume a point \((x,y)\) on the reflected triangle. A dilation centered at the origin with scale factor \(k\) transforms the point to \((kx,ky)\). By comparing the side - lengths of the blue triangle (after reflection) and the red triangle, we can find the scale factor. Suppose a side of the blue triangle (after reflection) has length \(l_1\) and the corresponding side of the red triangle has length \(l_2\). If we assume a vertex of the blue triangle (after reflection) is \((- 2,1)\) and the corresponding vertex of the red triangle is \((-2,2)\), using the dilation formula \((x,y)\to(kx,ky)\), substituting \(x=-2,y = 1,kx=-2,ky = 2\), we get \(k = 2\).

Answer:

\(x -\)axis; \(2\)