Question

fill in the blank 0 points extra credit one endpoint of a line segment is $(8,-1)$. the point $(5,-2)$ is one - third of the way from that endpoint to the other endpoint. find the other endpoint. what is the coordinate? (type your answer..., type your answer... )

Explanation:

Step1: Use the section formula

If a point \(P(x,y)\) divides the line segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\), then \(x=\frac{mx_2 + nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(A=(8,-1)\), \(P=(5,-2)\), and the ratio \(m:n = 1:2\) (since \(P\) is one - third of the way from \(A\) to \(B\), so \(AP:PB=1:2\)).
For the \(x\) - coordinate:
\(5=\frac{1\times x_2+2\times8}{1 + 2}\)
\(5=\frac{x_2 + 16}{3}\)
\(15=x_2 + 16\)
\(x_2=15 - 16=-1\)

Step2: For the \(y\) - coordinate:

\(-2=\frac{1\times y_2+2\times(-1)}{1 + 2}\)
\(-2=\frac{y_2-2}{3}\)
\(-6=y_2-2\)
\(y_2=-6 + 2=-4\)

Answer:

\((-1,-4)\)