Question

triangle abc and triangle def are graphed on the set of axes below. which sequence of transformations maps triangle abc into triangle def? a reflection over the x - axis followed by a reflection over the y - axis a 180^{circ} rotation about the origin followed by a reflection over the line y = x a 90^{circ} clockwise rotation about the origin followed by a reflection over the y - axis a 180 degree rotation about the origin, followed by a reflection across the x - axis

Explanation:

Step1: Analyze the position of the triangles

First, consider the general effect of a \(180^{\circ}\) rotation about the origin. For a point \((x,y)\) in the coordinate - plane, a \(180^{\circ}\) rotation about the origin transforms it to \((-x, -y)\).

Step2: Analyze the effect of reflection

A reflection across the \(x -\)axis transforms a point \((x,y)\) to \((x,-y)\).

Let's assume a general point \(A(x_1,y_1)\) in \(\triangle ABC\). A \(180^{\circ}\) rotation about the origin transforms \(A(x_1,y_1)\) to \(A'(-x_1,-y_1)\). Then a reflection across the \(x -\)axis transforms \(A'(-x_1,-y_1)\) to \(A''(-x_1,y_1)\)

We can also check by taking coordinates of vertices of \(\triangle ABC\) (assume \(A(x,y)\), \(B(m,n)\), \(C(p,q)\)) and applying the transformations one - by - one.

For a \(180^{\circ}\) rotation about the origin: \((x,y)\to(-x, -y)\), \((m,n)\to(-m,-n)\), \((p,q)\to(-p,-q)\)

For a reflection across the \(x -\)axis: \((-x, -y)\to(-x,y)\), \((-m,-n)\to(-m,n)\), \((-p,-q)\to(-p,q)\)

Answer:

\(180\) degree rotation about the origin, followed by a reflection across the \(x -\)axis.