Question

in line segment ab with endpoints a(9,11) and b(5, - 13), the point p is located $\frac{1}{4}$ of the way along ab from point b. determine the coordinates of point p.

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 $m:n$, the coordinates of $P$ are given by $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $P$ is located $\frac{1}{4}$ of the way along $AB$ from point $B$, so the ratio of $BP$ to $PA$ is $1:3$, i.e., $m = 1$ and $n = 3$, $x_1=9,y_1 = 11,x_2=5,y_2=-13$.

Step2: Calculate the $x$ - coordinate of $P$

$x=\frac{1\times5+3\times9}{1 + 3}=\frac{5 + 27}{4}=\frac{32}{4}=8$.

Step3: Calculate the $y$ - coordinate of $P$

$y=\frac{1\times(-13)+3\times11}{1 + 3}=\frac{-13 + 33}{4}=\frac{20}{4}=5$.

Answer:

$(8,5)$