To find the coordinates of [tex]\(Z\)[/tex] given that [tex]\(Y\)[/tex] is the midpoint of [tex]\(X\)[/tex] and [tex]\(Z\)[/tex] with [tex]\(X(-10, 9)\)[/tex] and [tex]\(Y(-4, 8)\)[/tex], we can use the midpoint formula.
The midpoint formula states that the coordinates of the midpoint [tex]\(M\)[/tex] of a line segment connecting two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are:
[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:
[tex]\[
Y = \left(\frac{X_1 + Z_1}{2}, \frac{X_2 + Z_2}{2}\right)
\][/tex]
Given:
[tex]\[
X = (-10, 9)
\][/tex]
[tex]\[
Y = (-4, 8)
\][/tex]
Let [tex]\(Z\)[/tex] be [tex]\((Z_1, Z_2)\)[/tex].
Using the x-coordinates:
[tex]\[
-4 = \frac{-10 + Z_1}{2}
\][/tex]
To find [tex]\(Z_1\)[/tex], solve the equation:
[tex]\[
-4 = \frac{-10 + Z_1}{2}
\][/tex]
[tex]\[
-4 \times 2 = -10 + Z_1
\][/tex]
[tex]\[
-8 = -10 + Z_1
\][/tex]
[tex]\[
Z_1 = -8 + 10
\][/tex]
[tex]\[
Z_1 = 2
\][/tex]
Using the y-coordinates:
[tex]\[
8 = \frac{9 + Z_2}{2}
\][/tex]
To find [tex]\(Z_2\)[/tex], solve the equation:
[tex]\[
8 = \frac{9 + Z_2}{2}
\][/tex]
[tex]\[
8 \times 2 = 9 + Z_2
\][/tex]
[tex]\[
16 = 9 + Z_2
\][/tex]
[tex]\[
Z_2 = 16 - 9
\][/tex]
[tex]\[
Z_2 = 7
\][/tex]
Therefore, the coordinates of [tex]\(Z\)[/tex] are:
[tex]\[
Z = (2, 7)
\][/tex]
