Answer:
[tex]Z(x,y) = (-18,7)[/tex]
Step-by-step explanation:
Given
[tex]X = (10,9)[/tex]
[tex]Y = (-4,8)[/tex]
Required
Determine the coordinates of Z
From the question, we understand that Y is the midpoint
And the midpoint is calculated as:
[tex](x,y) = (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]
In this case:
[tex]Y(x,y) = (-4,8)[/tex]
[tex]X(x_1,y_1) = (10,9)[/tex]
[tex]Z = (x_2,y_2)[/tex]
So: This gives
[tex](x,y) = (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]
[tex](-4,8) = (\frac{10+x_2}{2},\frac{9+y_2}{2})[/tex]
Multiply through by 2
[tex](-8,16) = (10 + x_2,9+y_2)[/tex]
By comparison:
[tex]-8 = 10 + x_2[/tex] and [tex]16 = 9 + y_2[/tex]
Solving for x2
[tex]-8 = 10 + x_2[/tex]
[tex]-8 - 10 = x_2[/tex]
[tex]-18 = x_2[/tex]
[tex]x_2 = -18[/tex]
Solving for y2
[tex]16 = 9 + y_2[/tex]
[tex]16 - 9 = y_2[/tex]
[tex]7 = y_2[/tex]
[tex]y_2 = 7[/tex]
Hence, the coordinates of Z is:
[tex]Z(x,y) = (-18,7)[/tex]
