Question

triangle gfh is the image of triangle gfh after a single transformation.
which similarity transformation maps triangle gfh?
\begin{array}{l}\bigcirc(x, y) \to(2 x, 2 y) \\bigcirc(x, y) \to(-2 x,-2 y) \\bigcirc(x, y) \to(1 / 2 x, 1 / 2 y) \\bigcirc(x, y) \to(-1 / 2 x,-1 / 2 y)end{array}

Explanation:

Step1: Analyze the transformation

Assume a general point \((x,y)\) in triangle \(GFH\). We need to check the effect of each transformation on the coordinates.
Let's take a sample point. Suppose \(H=( - 1,-2)\) (assuming the grid - based coordinates).

For the transformation \((x,y)\to(2x,2y)\)

If \(x=-1,y = - 2\), then the new coordinates are \((2\times(-1),2\times(-2))=(-2,-4)\)

For the transformation \((x,y)\to(-2x,-2y)\)

If \(x=-1,y=-2\), then the new coordinates are \((-2\times(-1),-2\times(-2))=(2,4)\)

For the transformation \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\)

If \(x = - 1,y=-2\), then the new coordinates are \((\frac{1}{2}\times(-1),\frac{1}{2}\times(-2))=(-\frac{1}{2},-1)\)

For the transformation \((x,y)\to(-\frac{1}{2}x,-\frac{1}{2}y)\)

If \(x=-1,y = - 2\), then the new coordinates are \((-\frac{1}{2}\times(-1),-\frac{1}{2}\times(-2))=(\frac{1}{2},1)\)

We can also consider the similarity transformation. A similarity transformation is a transformation that changes the size (scaling) and may change the orientation (reflection). The transformation \((x,y)\to(2x,2y)\) is a dilation (scaling) with a scale factor of \(2\) from the origin.

Answer:

\((x,y)\to(2x,2y)\)