Question

in the coordinate plane, the point ( x(1, -4) ) is translated to the point ( x(-1, -3) ). under the same translation, the points ( y(5, -7) ) and ( z(4, -6) ) are translated to ( y ) and ( z ), respectively. what are the coordinates of ( y ) and ( z )?

Explanation:

Step1: Find the translation vector

First, we find the translation vector by subtracting the coordinates of \( X \) from \( X' \). For the \( x \)-coordinate: \( -1 - 1 = -2 \). For the \( y \)-coordinate: \( -3 - (-4) = 1 \). So the translation vector is \( (-2, 1) \).

Step2: Translate point \( Y \) to \( Y' \)

For point \( Y(5, -7) \), we apply the translation vector. The new \( x \)-coordinate is \( 5 + (-2) = 3 \). The new \( y \)-coordinate is \( -7 + 1 = -6 \). So \( Y' = (3, -6) \).

Step3: Translate point \( Z \) to \( Z' \)

For point \( Z(4, -6) \), we apply the translation vector. The new \( x \)-coordinate is \( 4 + (-2) = 2 \). The new \( y \)-coordinate is \( -6 + 1 = -5 \). So \( Z' = (2, -5) \).

Answer:

\( Y'(3, -6) \), \( Z'(2, -5) \)