Question

example 1
find the coordinates of point x on
① point x on $overline{ab}$ is $\frac{1}{5}$ of the distance from a to b.
$a(-5, -5)$
$b(2, 9)$

Explanation:

Step1: Use the section - formula

The section - formula for a point \(X(x,y)\) that divides the line - segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\) is given by \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(m = 1\) and \(n=4\) (since the point \(X\) is \(\frac{1}{5}\) of the distance from \(A\) to \(B\), so the ratio of \(AX\) to \(XB\) is \(1:4\)), \(x_1=-5,y_1 = - 5,x_2=2,y_2 = 9\).

Step2: Calculate the \(x\) - coordinate of \(X\)

\[x=\frac{1\times2+4\times(-5)}{1 + 4}=\frac{2-20}{5}=\frac{-18}{5}=-3.6\]

Step3: Calculate the \(y\) - coordinate of \(X\)

\[y=\frac{1\times9+4\times(-5)}{1 + 4}=\frac{9-20}{5}=\frac{-11}{5}=-2.2\]

Answer:

\((-3.6,-2.2)\)