Respuesta :
The line segment connects the point (-5, 1) and (7, y), the segment has a length of 13 units. This is the distance between the points.
The equation for the distance of two points A and B, is:
[tex]\begin{gathered} \begin{cases}A=(x_A,y_A){} \\ B=(x_b.y_B){}\end{cases} \\ Distance=\sqrt{(x_A-x_B)^2+(y_A-y_B)} \end{gathered}[/tex]
Then, taking A = (-5, 1) and B = (7, y)
We can write:
[tex]13=\sqrt{(-5-7)^2+(1-y)^2}[/tex]Then, we can apply the square on both sides and solve the parentheses:
[tex]\begin{gathered} 13^2=(-12)^2+(1-2y+y^2) \\ 169=144+1+y^2-2y \\ y^2-2y-24=0 \\ \end{gathered}[/tex]Now we can apply the quadratic formula:
[tex]y_{1,2}=\frac{-(-2)\pm\sqrt{(-2)^2-4\cdot1\cdot(-24)}}{2\cdot1}[/tex][tex]y_{1,2}=\frac{2\pm\sqrt{4+96}}{2}=\frac{2\pm\sqrt{100}}{2}=\frac{2\pm10}{2}=1\pm5[/tex]The two solution are:
[tex]\begin{gathered} y_1=1+5=6 \\ y_2=1-5=-4 \end{gathered}[/tex]Then, the two answers are:
[tex]\begin{gathered} y=6 \\ y=-4 \end{gathered}[/tex]