Question

which best explains whether or not (\triangle abccong\triangle lmn)?
the figures are congruent because a (270^{circ}) rotation about the origin and then a reflection over the (x -)axis will map (\triangle abc) onto (\triangle lmn).
the figures are congruent because a 180 rotation about the origin and then a reflection over the (x -)axis will map (\triangle abc) onto (\triangle lmn).
the figures are not congruent because point (b) corresponds with point (n) and point (c) corresponds with point (m).
the figures are not congruent because there is no rigid transformation or combination of rigid transformations that will map (\triangle abc) onto (\triangle lmn).

Explanation:

Step1: Analyze the first option

A \(270^{\circ}\) rotation about the origin and then a reflection over the \(x -\)axis: Rigid transformations (rotations and reflections) preserve shape and size. But we need to check the mapping.

Step2: Analyze the second option

A \(180^{\circ}\) rotation about the origin and then a reflection over the \(x -\)axis: Let's assume \(A(-1,1)\), \(B(-5,1)\), \(C(-1,5)\) in \(\triangle ABC\). A \(180^{\circ}\) rotation about the origin \((x,y)\to(-x,-y)\) gives \(A'(1, - 1)\), \(B'(5,-1)\), \(C'(1,-5)\). Then reflection over \(x -\)axis \((x,y)\to(x,-y)\) gives \(A''(1,1)\), \(B''(5,1)\), \(C''(1,5)\) which does not match \(\triangle LMN\) (co - ordinates of \(L(1,-1)\), \(M(1,-4)\), \(N(5,-1)\))

Step3: Analyze the third option

If the figures were congruent, corresponding sides and angles would be equal. But the logic of non - congruence based on point correspondence is incorrect as we need to check all side lengths and angle measures (using distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\))

Step4: Analyze the fourth option

Calculate side lengths. For \(\triangle ABC\): \(AB=\sqrt{(-5 + 1)^2+(1 - 1)^2}=4\), \(AC=\sqrt{(-1+1)^2+(5 - 1)^2}=4\), \(BC=\sqrt{(-5 + 1)^2+(1 - 5)^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\). For \(\triangle LMN\): \(LM=\sqrt{(1 - 1)^2+(-4 + 1)^2}=3\), \(LN=\sqrt{(5 - 1)^2+(-1 + 1)^2}=4\), \(MN=\sqrt{(5 - 1)^2+(-1 + 4)^2}=5\). Since side lengths are not equal, there is no rigid transformation (rotation, reflection, translation) that can map \(\triangle ABC\) to \(\triangle LMN\)

Answer:

The figures are not congruent because there is no rigid transformation or combination of rigid transformations that will map \(\triangle ABC\) onto \(\triangle LMN\).