Question

which statement correctly identifies the line of reflection?
the triangles are reflected across the line ( y = -x ).
the triangles are reflected across the ( x - ) axis.
the triangles are reflected across the ( y - ) axis.
the triangles are reflected across the line ( y = x ).

Explanation:

Step1: Recall reflection properties

When reflecting a point \((x,y)\) across the line \(y = -x\), the transformation rule is \((x,y)\to(-y,-x)\).

Step2: Analyze the triangles

Let's assume a vertex of the upper - triangle (the non - purple one) is \((1,6)\). If we reflect \((1,6)\) across \(y=-x\), using the rule \((x,y)\to(-y,-x)\), we get \((-6,-1)\). A vertex of the lower (purple) triangle is near \((-6,-4)\) (approximate coordinate analysis based on the grid). When reflecting across \(y = -x\), for a right - angled triangle, the perpendicular sides (legs) of the right - triangle will be transformed according to the reflection rule.
The \(x\) - axis reflection rule is \((x,y)\to(x, - y)\), the \(y\) - axis reflection rule is \((x,y)\to(-x,y)\) and the line \(y = x\) reflection rule is \((x,y)\to(y,x)\). These rules do not map the vertices of the upper triangle to the vertices of the lower triangle as effectively as the \(y=-x\) reflection.

Answer:

The triangles are reflected across the line \(y=-x\)