Question

5 segment \\( \overline { a b } \\) is graphed on the grid below. point \\( c \\) is the image of point \\( b \\) after a tr
(a) map point \\( a \\) to point \\( d \\) using the same translation
and draw segment \\( \overline { c d } \\).
(b) state two relationships between the segments
\\( \overline { a b } \\) and \\( \overline { c d } \\).

1. \\( \overline { a b } \\) and \\( \overline { c d } \\) are
2. \\( \overline { a b } \\) and \\( \overline { c d } \\) are

(c) draw in segments \\( \overline { b c } \\) and \\( \overline { a d } \\). what type of figure is \\( a b c d \\)?

Explanation:

Step1: Find the translation rule

Let \(A(-4,1)\) and \(B(-1,6)\), \(C(4,3)\). The translation from \(B\) to \(C\) is \((x + 5,y - 3)\) (because \(4-(-1)=5\) and \(3 - 6=-3\)).

Step2: Apply the translation rule to point \(A\)

For point \(A(-4,1)\), after translation \(x=-4 + 5=1\), \(y=1-3=-2\). So point \(D(1,-2)\)

Step3: For part (b)

1. Since translation preserves length, \(\overline{AB}\) and \(\overline{CD}\) are congruent (\(AB = CD\)).
2. Since translation preserves direction, \(\overline{AB}\) and \(\overline{CD}\) are parallel (\(AB\parallel CD\)).

Step4: For part (c)

After drawing \(\overline{BC}\) and \(\overline{AD}\), since \(AB\parallel CD\) and \(AB = CD\), \(BC\parallel AD\) and \(BC = AD\) (by translation properties), the figure \(ABCD\) is a parallelogram.

Answer:

(a) Plot \(D(1,-2)\) and draw \(\overline{CD}\).
(b) 1. congruent 2. parallel
(c) parallelogram