Question

which best explains whether or not \\( \triangle a b c \cong \triangle l m n \\)?
the figures are congruent because a \\( 270 ^ { \circ } \\) rotation about the origin and then a reflection over the \\( x \\) - axis will map \\( \triangle a b c \\) onto \\( \triangle l m n \\).
the figures are congruent because a 180 rotation about the origin and then a reflection over the \\( x \\) - axis will map \\( \triangle a b c \\) onto \\( \triangle l m n \\).
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 a b c \\) onto \\( \triangle l m n \\).

Explanation:

Step1: Analyze the side lengths

Let's assume the grid has unit - length squares.
For \(\triangle ABC\), using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
If \(A(-1,1)\), \(B(-5,1)\), \(C(-2,5)\)
\(AB=\sqrt{(-1 + 5)^2+(1 - 1)^2}=\sqrt{16}=4\), \(AC=\sqrt{(-1+2)^2+(1 - 5)^2}=\sqrt{1 + 16}=\sqrt{17}\), \(BC=\sqrt{(-5 + 2)^2+(1 - 5)^2}=\sqrt{9 + 16}=5\)
For \(\triangle LMN\), if \(L(1,-1)\), \(M(1,-4)\), \(N(5,-2)\)
\(LM=\sqrt{(1 - 1)^2+(-1+4)^2}=3\), \(LN=\sqrt{(1 - 5)^2+(-1 + 2)^2}=\sqrt{16 + 1}=\sqrt{17}\), \(MN=\sqrt{(5 - 1)^2+(-2 + 4)^2}=\sqrt{16+4}=\sqrt{20}\)

Step2: Check congruence criteria

Congruent triangles must satisfy \(SSS\) (side - side - side), \(SAS\) (side - angle - side), \(ASA\) (angle - side - angle), \(AAS\) (angle - angle - side) or \(HL\) (hypotenuse - leg for right - angled triangles) criteria.
Since \(AB
eq LM\) (\(4
eq3\)), \(BC
eq MN\) (\(5
eq\sqrt{20}\)), there is no rigid transformation (translation, rotation, reflection) that can map \(\triangle ABC\) onto \(\triangle LMN\) as rigid transformations preserve side - lengths and angles.

Answer:

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