Question

persevere point p is located on the segment between point a(1,4) and point d(7,13). the distance from a to p is twice the distance from p to d. what are the coordinates of point p?

Explanation:

Step1: Find the ratio of AP to AD

Since the distance from $A$ to $P$ is twice the distance from $P$ to $D$, then $AP:PD = 2:1$, and $AP:AD=2:(2 + 1)=2:3$.

Step2: Use the section - formula for x - coordinate

The section formula for the x - coordinate of a point $P(x,y)$ that divides the line segment joining $A(x_1,y_1)$ and $D(x_2,y_2)$ in the ratio $m:n$ is $x=\frac{mx_2+nx_1}{m + n}$. Here, $x_1 = 1$, $x_2=7$, $m = 2$, $n = 1$. So $x=\frac{2\times7+1\times1}{2 + 1}=\frac{14 + 1}{3}=\frac{15}{3}=5$.

Step3: Use the section - formula for y - coordinate

The section formula for the y - coordinate is $y=\frac{my_2+ny_1}{m + n}$. Here, $y_1 = 4$, $y_2 = 13$, $m = 2$, $n = 1$. So $y=\frac{2\times13+1\times4}{2+1}=\frac{26 + 4}{3}=\frac{30}{3}=10$.

Answer:

$(5,10)$