Given points (0, 0) and (3, 6), find the point which partitions the segment into 3:2
Please explain?
Point P partitions segment AB in a m:n ratio.
Use Section rormula.
If point P partitions AB in a m:n ratio, then the coordinates of point P are:
x coordinate = ( m ∙ x2 + n ∙ x1 ) / ( m + n )
y coordinate = ( m ∙ y2 + n ∙ y1 ) / ( m + n )
In this case:
x1 = 0 , x2 = 3 , y1 = 0 , y2 = 6 , m = 3 , n = 2
So:
x = ( m ∙ x2 + n ∙ x1 ) / ( m + n )
x = ( 3 ∙ 3 + 2 ∙ 0 ) / ( 3 + 2 )
x = 9 / 5
y = ( m ∙ y2 + n ∙ y1 ) / ( m + n )
y = ( 3 ∙ 6 + 2 ∙ 0 ) / ( 3 + 2 )
y = 18 / 5
P ( 9 / 5 , 18 / 5 )
To find the point that partitions a line segment into a ratio of 3:2, you need to use the concept of section formula or the internal division formula. This formula divides the line segment into two parts based on the given ratio.
Let's assume the given points as A(0, 0) and B(3, 6). The ratio 3:2 indicates that the line segment is divided into three parts and two parts, respectively.
To find the point that partitions the line segment into the given ratio, you can follow these steps:
Step 1: Determine the total length of the line segment.
The distance between points A(0, 0) and B(3, 6) can be calculated using the distance formula, which is:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, the formula becomes:
Distance = sqrt((3 - 0)^2 + (6 - 0)^2)
= sqrt(9 + 36)
= sqrt(45)
So, the total length of the line segment AB is sqrt(45).
Step 2: Divide the total length by the sum of the ratio parts.
To find the length of one part (3:2), add the ratio parts and divide the total length by the sum of the parts:
Length of one part = Total length / (3 + 2)
= sqrt(45) / 5
Step 3: Find the coordinates of the point dividing the line segment.
Using the ratio and the calculated length of one part, you can find the coordinates (x, y) of the point.
Let's assume the coordinates of the dividing point to be (x, y).
To calculate x coordinate, use the formula:
x = ( (x1 * n2) + (x2 * n1) ) / (n1 + n2)
To calculate y coordinate, use the formula:
y = ( (y1 * n2) + (y2 * n1) ) / (n1 + n2)
In this case, n1 = 3 (the number of parts for the first point of the ratio) and n2 = 2 (the number of parts for the second point of the ratio).
So, the coordinates of the point that partitions the line segment AB into a ratio of 3:2 are:
x = ( (0 * 2) + (3 * 3) ) / (3 + 2)
= 9 / 5
y = ( (0 * 2) + (6 * 3) ) / (3 + 2)
= 18 / 5
Hence, the point that partitions the line segment AB into a ratio of 3:2 is (9/5, 18/5).