Question

nate wants to visit his friend mac before going to the park. nates house is located at (-4, 6), while the park is located at (12, 4). find the location of macs house if it is 1/2 of the distance from nates house to the park. the location of macs house is ( , ).

Explanation:

Step1: Use section - formula

Let the coordinates of Nate's house be $A(-4,6)$ and the coordinates of the park be $B(12,4)$. 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 section formula is given by $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $m = 1$ and $n = 1$ (since Mac's house is the mid - point, i.e., it divides the line segment from Nate's house to the park in the ratio $1:1$).

Step2: Calculate the x - coordinate

$x=\frac{1\times12+1\times(-4)}{1 + 1}=\frac{12 - 4}{2}=\frac{8}{2}=4$.

Step3: Calculate the y - coordinate

$y=\frac{1\times4+1\times6}{1 + 1}=\frac{4 + 6}{2}=\frac{10}{2}=5$.

Answer:

$(4,5)$