Question

eric says that \\( \overline { a b } \\) was transformed to \\( \overline { c d } \\) using a combination of only rotations, reflections, and translations.
which best explains whether or not eric is correct?
a eric is correct because \\( \overline { a b } \\) was rotated and then translated.
b. eric is correct because \\( \overline { a b } \\) was reflected over the y - axis and then rotated and translated.
c. eric is not correct because the lengths of the segments are not the same.
d eric is not correct because the orientations of the segments are not the same.

Explanation:

Step1: Calculate the length of \( \overline{AB} \)

Using the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \) for points \( A(1,4) \) and \( B(4,4) \).
\( d_{AB}=\sqrt{(4 - 1)^2+(4 - 4)^2}=\sqrt{3^2+0^2}=3 \)

Step2: Calculate the length of \( \overline{CD} \)

For points \( C(-4,-1) \) and \( D(-2,-4) \).
\( d_{CD}=\sqrt{(-2+4)^2+(-4 + 1)^2}=\sqrt{2^2+(-3)^2}=\sqrt{4 + 9}=\sqrt{13}\)
Since \( 3
eq\sqrt{13} \), the lengths of \( \overline{AB} \) and \( \overline{CD} \) are not equal. Rotations, reflections, and translations are rigid transformations that preserve the length of a segment.

Answer:

C. Eric is not correct because the lengths of the segments are not the same.