Answer:
[tex]x =12\ or\ -8[/tex]
Step-by-step explanation:
Given
[tex](x_1,y_1) = (6,5)[/tex]
[tex](x_2,y_2) = (8,2)[/tex]
[tex](x_3,y_3) = (x,11)[/tex]
[tex]Area = 15[/tex]
Required
Find x
The area is calculated as thus:
[tex]Area = \frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|[/tex]
Substitute values
[tex]15 = \frac{1}{2}|6*(2 - 11) + 8*(11 - 5) + x(5 - 2)|[/tex]
[tex]15 = \frac{1}{2}|6*(-9) + 8*(6) + x(3)|[/tex]
[tex]15 = \frac{1}{2}|-54 + 48 + 3x|[/tex]
[tex]15 = \frac{1}{2}|-6 + 3x|[/tex]
Multiply through by 2
[tex]2 * 15 = \frac{1}{2}|-6 + 3x| * 2[/tex]
[tex]30 = |-6 + 3x|[/tex]
Remove absolute sign
[tex]30 = -6 + 3x\ or\ -30 = -6 + 3x[/tex]
Add 6 to both sides
[tex]36 = 3x\ or\ -24 = 3x[/tex]
Divide by 3
[tex]12 = x\ or\ -8 = x[/tex]
[tex]x =12\ or\ -8[/tex]
So, the coordinates are (-8, 11) or (12,11)
