Question

Given the directed segment AB such that A(5,0) and B(-3,-4). Find the coordinates of point P which partitions the segment in a ratio of 3:2

Answer 1

`P=\left((1)/(5),-(12)/(5)\right)`

Explanation:

We can use the internal section formula to find the coordinates of point P which divides the directed segment AB in the ratio of 3 : 2.

Internal Section formula

`P(x,y)=\left((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n)\right)`

where:

• (x₁ ,y₁) are the coordinates of point A.
• (x₂, y₂) are the coordinates of the point B.
• m and n are the ratio values in which P divides the line internally.

Given A (5, 0) and B (-3, -4) and the ratio 3 : 2:

• x₁ = 5
• y₁ = 0
• x₂ = -3
• y₂ = -4
• m = 3
• n = 2

Substitute the values into the formula:

`\implies P(x,y)=\left((3 \cdot (-3)+2 \cdot 5)/(3+2),(3 \cdot (-4)+2 \cdot 0)/(3+2)\right)`

`\implies P(x,y)=\left((-9+10)/(5),(-12+0)/(5)\right)`

`\implies P(x,y)=\left((1)/(5),-(12)/(5)\right)`

Therefore, the coordinates of point P which partitions segment AB in a ratio of 3 : 2 are:

`\bullet\quad\left((1)/(5),-(12)/(5)\right)`

Note: The segment to be partitioned is a directed segment. Direction is important in a directed segment; it has a clear starting point and ending point. The starting point is the first letter of the segment and the ending point is the last letter of the segment. Therefore, if the directed segment AB is partitioned in a ratio 3 : 2 this means that point P is 3/5ths along the line from point A to point B.