Question

the coordinates of one endpoint of a line segment are (-5,1) parentheses. The line segment is 13 units long. Find the value of "y" if the other end point is ( 7,y)

Answer 1

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:

$\begin{gathered} \begin{cases}A=(x_A,y_A){} \\ B=(x_b.y_B){}\end{cases} \\ Distance=\sqrt{(x_A-x_B)^2+(y_A-y_B)} \end{gathered}$

Then, taking A = (-5, 1) and B = (7, y)

We can write:

$13=\sqrt{(-5-7)^2+(1-y)^2}$

Then, we can apply the square on both sides and solve the parentheses:

$\begin{gathered} 13^2=(-12)^2+(1-2y+y^2) \\ 169=144+1+y^2-2y \\ y^2-2y-24=0 \\ \end{gathered}$

Now we can apply the quadratic formula:

$y_{1,2}=\frac{-(-2)\pm\sqrt{(-2)^2-4\cdot1\cdot(-24)}}{2\cdot1}$$y_{1,2}=\frac{2\pm\sqrt{4+96}}{2}=\frac{2\pm\sqrt{100}}{2}=\frac{2\pm10}{2}=1\pm5$

The two solution are:

$\begin{gathered} y_1=1+5=6 \\ y_2=1-5=-4 \end{gathered}$

Then, the two answers are:

$\begin{gathered} y=6 \\ y=-4 \end{gathered}$