Question

NO LINKS!! URGENT HELP PLEASE!!!

Find the coordinates of point P along the directed line segment AB that partitions it so that the ratio of AP to PB is 2:1. Write your answer in the form (a,b).

Answer 1

(5, 6)

Explanation:

If the line segment AB is partitioned so that the ratio of AP to PB is 2 : 1, then point P is two-thirds of the way along AB.

To find the location of point P on the coordinate plane, we can use the Section Formula for Internal Division:

`\boxed{\begin{minipage}{8.1 cm}\underline{Section Formula - Internal Division}\\\\\\$P(x,y)=\left((nx_1+mx_2)/(m+n),(ny_1+my_2)/(m+n)\right)$\\\\\\where:\\\phantom{ww} $\bullet$ $\overline{AB}$ is the directed line segment.\\\phantom{ww} $\bullet$ $A(x_1, y_1)$ and $B(x_2, y_2)$ are the endpoints.\\\phantom{ww} $\bullet$ Point $P$ divides the segment in the ratio $m : n$.\\ \end{minipage}}`

Point A is located at (3, 2) and point B is located at point (6, 8).

The direction of the line segment AB is point A to point B. Therefore, A is the first endpoint (x₁, y₁) and B is the second endpoint (x₂, y₂).

The given ratio of AP to PB is 2 : 1, so m : n = 2 : 1.

Therefore, the values to substitute into the formula are:

• x₁, = 3
• y₁ = 2
• x₂ = 6
• y₂ = 8
• m = 2
• n = 1

Substitute the values into the formula:

`\begin{aligned}P(x,y)&=\left((nx_1+mx_2)/(m+n),(ny_1+my_2)/(m+n)\right)\\\\&=\left((1 \cdot 3+2 \cdot 6)/(2+1),(1 \cdot 2+2 \cdot 8)/(2+1)\right)\\\\&=\left((3+12)/(3),(2+16)/(3)\right)\\\\&=\left((15)/(3),(18)/(3)\right)\\\\&=\left(5,6\right)\end{aligned}`

Therefore, the location of point P on the coordinate plane is (5, 6).