To find the coordinates of point [tex]\( Z \)[/tex] given that [tex]\( Y \)[/tex] is the midpoint of the line segment [tex]\(\overline{XZ}\)[/tex], where [tex]\( X \)[/tex] has coordinates [tex]\((-10, 9)\)[/tex] and [tex]\( Y \)[/tex] has coordinates [tex]\((-4, 8)\)[/tex], you can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint [tex]\( M \)[/tex] of a line segment with endpoints [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] are given by:
[tex]\[ M\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \][/tex]
In this problem, [tex]\( Y \)[/tex] is the midpoint, so we know:
[tex]\[ Y = \left( \frac{X_x + Z_x}{2}, \frac{X_y + Z_y}{2} \right) \][/tex]
Given:
[tex]\[ X(-10, 9) \][/tex]
[tex]\[ Y(-4, 8) \][/tex]
We can set up the following equations based on the midpoint formula:
1. For the [tex]\( x \)[/tex]-coordinate:
[tex]\[ -4 = \frac{-10 + Z_x}{2} \][/tex]
2. For the [tex]\( y \)[/tex]-coordinate:
[tex]\[ 8 = \frac{9 + Z_y}{2} \][/tex]
To solve for [tex]\( Z_x \)[/tex] and [tex]\( Z_y \)[/tex]:
For the [tex]\( x \)[/tex]-coordinate:
[tex]\[ -4 = \frac{-10 + Z_x}{2} \][/tex]
Multiply both sides by 2:
[tex]\[ -8 = -10 + Z_x \][/tex]
Add 10 to both sides:
[tex]\[ Z_x = 2 \][/tex]
For the [tex]\( y \)[/tex]-coordinate:
[tex]\[ 8 = \frac{9 + Z_y}{2} \][/tex]
Multiply both sides by 2:
[tex]\[ 16 = 9 + Z_y \][/tex]
Subtract 9 from both sides:
[tex]\[ Z_y = 7 \][/tex]
Therefore, the coordinates of point [tex]\( Z \)[/tex] are [tex]\( (2, 7) \)[/tex].
