Question

segment pm is graphed on the coordinate plane such that point p is located at (2,-5). point a is graphed exactly 1/4 of the distance from p to m and is located at (4,-1). what is the value of the y - coordinate of point m?

Explanation:

Step1: Use section - formula concept

Let the coordinates of point $M$ be $(x_M,y_M)$. The section - formula for a point $A(x_A,y_A)$ that divides the line - segment joining $P(x_P,y_P)$ and $M(x_M,y_M)$ in the ratio $m:n$ is given by $x_A=\frac{mx_M+nx_P}{m + n}$ and $y_A=\frac{my_M+ny_P}{m + n}$. Here, $m = 1$, $n = 3$ (since $A$ divides $PM$ in the ratio $1:3$), $x_P = 2$, $y_P=-5$, $x_A = 4$, and $y_A=-1$.
We will use the $y$ - coordinate part of the section formula $y_A=\frac{my_M+ny_P}{m + n}$.

Step2: Substitute the values into the formula

Substitute $m = 1$, $n = 3$, $y_P=-5$, and $y_A=-1$ into $y_A=\frac{my_M+ny_P}{m + n}$.
We get $-1=\frac{1\times y_M+3\times(-5)}{1 + 3}$.

Step3: Solve the resulting equation for $y_M$

First, simplify the right - hand side of the equation: $-1=\frac{y_M-15}{4}$.
Multiply both sides of the equation by $4$: $-1\times4=y_M - 15$.
So, $-4=y_M-15$.
Add $15$ to both sides of the equation: $y_M=-4 + 15$.
$y_M = 11$.

Answer:

$11$