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=√((x_A-x_B)^2+(y_A-y_B)) \end{gathered}`

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

We can write:

`13=√((-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)=(-(-2)\pm√((-2)^2-4\cdot1\cdot(-24)))/(2\cdot1)`
`y_(1,2)=(2\pm√(4+96))/(2)=(2\pm√(100))/(2)=(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}`