To determine why shape 1 and shape 2 are not congruent, we first need to understand what congruence means in geometry. Congruent shapes are those that can be completely matched through a series of rigid transformations, which include translations, rotations, and reflections. These transformations do not alter the size or shape of the figures.
Let's refer to the coordinates provided for the vertices of the shapes:
Shape 1:
A(4, 8), B(6, 8), C(8, 6), D(8, 4), E(6, 2), F(4, 2), G(2, 6)
Shape 2:
A'(12, 14), B'(14, 14), C'(16, 12), D'(16, 10), E'(14, 8), F'(12, 8), G'(10, 10), H'(10, 12)
Given the coordinates, we observe that shape 1 is a heptagon (7-pointed shape) whereas shape 2 is an octagon (8-pointed shape). Since shape 1 and shape 2 have a different number of sides and vertices, identifying a congruence through rigid transformations is impossible because:
- They do not have the same number of corresponding sides and angles.
- Dilation changes size, not shape, and does not alter the number of vertices.
Therefore, the sentence that best explains why shape 1 and shape 2 are not congruent is:
B. There is no sequence of rigid transformations that will map shape 1 onto shape 2.
Which sentence best explains why shape 1 and shape 2 are not congruent?
A(4,8) B(6,8) C(8,6) D(8,4) E(6,2) F(4,2) G(2,6)
A'(12,14) B'(14,14) C'(16,12) D'(16,10) E'(14,8) F'(12,8) G'(10,10) H'(10,12)
A. Not all corresponding pairs of sides on the two shapes are parallel.
B. There is no sequence of rigid transformations that will map shape 1 onto shape 2.
C. Shape 1 cannot be mapped onto shape 2 using a reflection.
D. Not all corresponding pairs of sides on the two shapes are perpendicular.
E. Shape 1 cannot be mapped onto shape 2 using a dilation.
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