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A sign is being created using two trapezoids where trapezoid [tex]\( E^{\prime} F^{\prime} G^{\prime} H^{\prime} \)[/tex] is the translation of trapezoid [tex]\( E F G H \)[/tex]. The table of translations is below:

\begin{tabular}{|c|c|}
\hline
Trapezoid [tex]\(E F G H\)[/tex] & Trapezoid [tex]\(E^{\prime} F^{\prime} G^{\prime} H^{\prime}\)[/tex] \\
\hline
[tex]$E(-1, 4)$[/tex] & [tex]$E^{\prime}(2, 2)$[/tex] \\
\hline
[tex]$F$[/tex] & [tex]$F^{\prime}(4, 2)$[/tex] \\
\hline
[tex]$G(2, 1)$[/tex] & [tex]$G^{\prime}(5, -1)$[/tex] \\
\hline
[tex]$H(-3, 1)$[/tex] & [tex]$H^{\prime}$[/tex] \\
\hline
\end{tabular}

Find the coordinates of point [tex]\( H^{\prime} \)[/tex]:

A. [tex]\(H^{\prime}(-6, 3)\)[/tex]
B. [tex]\(H^{\prime}(-6, -1)\)[/tex]
C. [tex]\(H^{\prime}(0, -1)\)[/tex]
D. [tex]\(H^{\prime}(0, 1)\)[/tex]



Answer :

GinnyAnswer
To solve this problem, we need to determine the translation vector used to move from trapezoid [tex]\( EFGH \)[/tex] to [tex]\( E'F'G'H' \)[/tex].

First, observe the translation of point [tex]\( E \)[/tex] to [tex]\( E' \)[/tex]:
- Point [tex]\( E \)[/tex] has coordinates [tex]\((-1, 4)\)[/tex].
- Point [tex]\( E' \)[/tex] has coordinates [tex]\((2, 2)\)[/tex].

To determine the translation vector that moves [tex]\( E \)[/tex] to [tex]\( E' \)[/tex]:
1. Calculate the change in the x-coordinate:
[tex]\[ \Delta x = E'_{\text{x}} - E_{\text{x}} = 2 - (-1) = 3 \][/tex]
2. Calculate the change in the y-coordinate:
[tex]\[ \Delta y = E'_{\text{y}} - E_{\text{y}} = 2 - 4 = -2 \][/tex]

Thus, the translation vector is [tex]\((\Delta x, \Delta y) = (3, -2)\)[/tex].

Next, we apply the same translation vector to point [tex]\( H \)[/tex]:
- Point [tex]\( H \)[/tex] has coordinates [tex]\((-3, 1)\)[/tex].

Applying the translation vector [tex]\((3, -2)\)[/tex] to [tex]\( H \)[/tex]:
[tex]\[ H'_{\text{x}} = H_{\text{x}} + 3 = -3 + 3 = 0 \][/tex]
[tex]\[ H'_{\text{y}} = H_{\text{y}} - 2 = 1 - 2 = -1 \][/tex]

Therefore, the coordinates of point [tex]\( H' \)[/tex] are [tex]\((0, -1)\)[/tex]. So, the correct answer is:
[tex]\[ H' = (0, -1) \][/tex]

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