Quadrilateral KLMN has vertices K(-2,-2), L(1,1), M(0,4), and N(-3,5). It is first translated by (x + 2,y - 1) and then translated by (x - 3,y + 4). When a figure is translated twice, a double prime symbol is used. Find the coordinates of K" L" M" N" after both translations.

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To find the coordinates of K", L", M", N" after both translations, we need to apply the translations one after another to the original coordinates of KLMN.

Translation 1: (x + 2, y - 1)
Translation 2: (x - 3, y + 4)

Let's start with the original coordinates of KLMN:

K(-2, -2)
L(1, 1)
M(0, 4)
N(-3, 5)

Applying Translation 1:
K' = K + (2, -1)
L' = L + (2, -1)
M' = M + (2, -1)
N' = N + (2, -1)

K' = (-2, -2) + (2, -1) = (0, -3)
L' = (1, 1) + (2, -1) = (3, 0)
M' = (0, 4) + (2, -1) = (2, 3)
N' = (-3, 5) + (2, -1) = (-1, 4)

Now, let's apply Translation 2 to the coordinates we obtained:

K" = K' + (-3, 4)
L" = L' + (-3, 4)
M" = M' + (-3, 4)
N" = N' + (-3, 4)

K" = (0, -3) + (-3, 4) = (-3, 1)
L" = (3, 0) + (-3, 4) = (0, 4)
M" = (2, 3) + (-3, 4) = (-1, 7)
N" = (-1, 4) + (-3, 4) = (-4, 8)

Therefore, the coordinates of K", L", M", N" after both translations are:
K"(-3, 1)
L"(0, 4)
M"(-1, 7)
N"(-4, 8)

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K(-2,-2) ----> K'(0,-3) ----> K''(-3, 1)

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