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:

1. First, recall the properties of rigid transformations (rotations, reflections, translations) which preserve congruence.
2. Analyze the coordinates: Let's find coordinates of \( \triangle ABC \) and \( \triangle LMN \).

• \( A(-1,1) \), \( B(-4,1) \), \( C(-1,5) \)
• \( L(1,-1) \), \( M(1,-4) \), \( N(5,-1) \)

1. A \( 270^\circ \) rotation about the origin: The rule for \( 270^\circ \) rotation (\( (x,y) \to (y, -x) \)):

• For \( A(-1,1) \): \( (1, 1) \)? Wait, no, correct rule for \( 270^\circ \) counterclockwise is \( (x,y) \to (y, -x) \), clockwise is \( (x,y) \to (-y, x) \). Let's use clockwise \( 270^\circ \) (same as counterclockwise \( 90^\circ \) reverse). Wait, maybe better to check the transformation:
• After \( 270^\circ \) rotation (let's do counterclockwise \( 270^\circ \) which is equivalent to clockwise \( 90^\circ \)): \( (x,y) \to (y, -x) \)
• \( A(-1,1) \to (1, 1) \)? No, wait \( 270^\circ \) counterclockwise: \( (x,y) \to (y, -x) \). So \( A(-1,1) \to (1, 1) \)? Wait, no, \( x=-1, y=1 \), so \( y=1, -x = 1 \), so \( (1,1) \). Then \( B(-4,1) \to (1, 4) \). \( C(-1,5) \to (5, 1) \). Then reflection over x-axis: \( (x,y) \to (x, -y) \). So \( (1,1) \to (1,-1) \) (which is L), \( (1,4) \to (1,-4) \) (which is M), \( (5,1) \to (5,-1) \) (which is N). So this transformation maps \( \triangle ABC \) to \( \triangle LMN \).
• Check other options:
• 180 rotation: \( (x,y) \to (-x,-y) \). \( A(-1,1) \to (1,-1) \), \( B(-4,1) \to (4,-1) \), \( C(-1,5) \to (1,-5) \). Then reflection over x-axis: \( (x,y) \to (x,-y) \), which would not match LMN.
• The third option is wrong because correspondence of points doesn't mean non - congruent; rigid transformations can map them.
• The fourth option is wrong as we found a rigid transformation (rotation + reflection) that maps them.

Answer:

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 \( \triangle LMN \).