Question

1. triangles dwf, dwf and dwf are shown on the coordinate grid. what sequence of transformations maps triangle dwf onto triangle dwf and then onto triangle dwf? dwf→dwf dwf→dwf

Explanation:

Step 1: Determine the transformation from \(DWF\) to \(D'W'F'\)

Looking at the coordinates of the vertices of \(DWF\) and \(D'W'F'\), we can observe that the \(x\) - coordinates and \(y\) - coordinates of the vertices of \(DWF\) change their signs to get the vertices of \(D'W'F'\).
If a point \((x,y)\) is transformed to \((-x,-y)\), the transformation is a rotation of \(180^{\circ}\) about the origin. Let \((x,y)\) be a vertex of \(\triangle DWF\). After rotation of \(180^{\circ}\) about the origin, using the rotation formula \((x,y)\to(-x,-y)\), we get the vertices of \(\triangle D'W'F'\).

Step 2: Determine the transformation from \(D'W'F'\) to \(D''W''F''\)

Looking at the coordinates of the vertices of \(D'W'F'\) and \(D''W''F''\). If we consider a point \((x,y)\) of \(\triangle D'W'F'\) and a point \((x + a,y)\) of \(\triangle D''W''F''\).
We can see that the \(y\) - coordinates of the vertices remain the same and the \(x\) - coordinates of the vertices of \(D'W'F'\) are increased by \(6\). The transformation is a translation. The translation rule is \((x,y)\to(x + 6,y)\) (a translation of \(6\) units to the right).

Answer:

\(DWF\to D'W'F'\): Rotation of \(180^{\circ}\) about the origin.
\(D'W'F'\to D''W''F''\): Translation \(6\) units to the right.