Question

which best explains whether or not δabc ≅ δlmn? the figures are congruent because a 270° rotation about the origin and then a reflection over the x-axis will map δabc onto δlmn. the figures are congruent because a 180 rotation about the origin and then a reflection over the x-axis will map δabc onto δlmn. 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 δabc onto δlmn.

Explanation:

Step1: Analyze rotation and reflection effects

A \(270^{\circ}\) rotation about the origin \((x,y)\to(y, - x)\) and then reflection over \(x -\)axis \((y,-x)\to(y,x)\).
For \(\triangle ABC\) with \(A(-1,1)\), \(B(-5,1)\), \(C(-1,5)\). After \(270^{\circ}\) rotation: \(A(1,1)\), \(B(1,5)\), \(C(5,1)\). After reflection over \(x -\)axis: \(A(1, - 1)\), \(B(1, - 5)\), \(C(5, - 1)
eq\triangle LMN\) ( \(L(1,-1)\), \(M(1,-4)\), \(N(5,-1)\)).
A \(180^{\circ}\) rotation about the origin \((x,y)\to(-x,-y)\). For \(\triangle ABC\): \(A(1,-1)\), \(B(5,-1)\), \(C(1,-5)\). After reflection over \(x -\)axis: \(A(1,1)\), \(B(5,1)\), \(C(1,5)
eq\triangle LMN\).
Since congruent figures can be mapped via rigid - transformations (rotation, reflection, translation). If we check side lengths:
Length of \(AB=\vert-1+5\vert = 4\), \(AC=\vert1 - 5\vert=4\), \(BC=\sqrt{(-5 + 1)^{2}+(1 - 5)^{2}}=\sqrt{16 + 16}=4\sqrt{2}\).
Length of \(LM=\vert-1+4\vert = 3\), \(LN=\vert1 - 5\vert=4\), \(MN=\sqrt{(5 - 1)^{2}+(-1+4)^{2}}=\sqrt{16 + 9}=5\).
Since side lengths are not equal ( \(AB
eq LM\)), there is no rigid transformation to 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 \(\triangle LMN\).