Question

20
which sentence best explains why shape 1 and shape 2 are not congruent?
not all corresponding pairs of sides on the two shapes are parallel.
shape 1 cannot be mapped onto shape 2 using a dilation.
not all corresponding pairs of sides on the two shapes are perpendicular.
there is no sequence of rigid transformations that will map shape 1 onto shape 2.
shape 1 cannot be mapped onto shape 2 using a reflection.

Explanation:

Analyze the shapes

Using the Rigid Transformations knowledge point

• Shape 1 is a heptagon (7-sided polygon) with vertices at \((2,6)\), \((4,8)\), \((6,8)\), \((8,6)\), \((8,4)\), \((6,2)\), and \((4,2)\).
• Shape 2 is an octagon (8-sided polygon) with vertices at \((10,10)\), \((10,12)\), \((12,14)\), \((14,14)\), \((16,12)\), \((16,10)\), \((14,8)\), and \((12,8)\).

Apply congruence definition

• Congruent figures must have the same size and shape.
• Mathematically, two figures are congruent if and only if there exists a sequence of rigid transformations (translations, rotations, and reflections) that maps one figure onto the other.
• Since Shape 1 has 7 sides and Shape 2 has 8 sides, they are not congruent.
• Therefore, no sequence of rigid transformations can map Shape 1 onto Shape 2.

Evaluate the options

• Option 1: Parallel sides are not required for congruence.
• Option 2: Dilations change size, which is not used to show congruence.
• Option 3: Perpendicular sides are not required for congruence.
• Option 4: Correctly states that no sequence of rigid transformations can map Shape 1 onto Shape 2.
• Option 5: A single reflection is not the only way to show congruence.

Answer:

• Not all corresponding pairs of sides on the two shapes are parallel.
• Shape 1 cannot be mapped onto shape 2 using a dilation.
• Not all corresponding pairs of sides on the two shapes are perpendicular.
• There is no sequence of rigid transformations that will map shape 1 onto shape 2. (Correct answer)
• Shape 1 cannot be mapped onto shape 2 using a reflection.