Question

Point A has coordinates (2,4). Point B has coordinates of (10,12). Find the coordinates of point P that partition AB in the ratio 1:3.

a) (3,5)
b) (4,6)
c) (5,7)
d) (6,8)

Answer 1

The coordinates of point P that partition AB in the ratio 1:3 are (6,8)

So, the correct answer is d) (6,8).

Step-by-step explanation:

To find the coordinates of point P, which partitions AB in the ratio 1:3, we use the section formula. Let P have coordinates (x,y). The coordinates are given by:

`\[ x = ((3 \cdot 10) + (1 \cdot 2))/(4) = 6 \]`

`\[ y = ((3 \cdot 12) + (1 \cdot 4))/(4) = 8 \]`

So, the correct answer is d) (6,8).

The section formula is a valuable tool for finding the coordinates of a point that partitions a line segment in a given ratio. Understanding how to apply this formula is essential in geometry and coordinate geometry, allowing for precise calculations of points on a line segment.

Answer 2

The coordinates of point P that partition AB in the ratio 1:3 are (4,6). This aligns with option b) (4,6). This is found using the section formula for dividing a line segment. Therefore the correct answer is option b.

Step-by-step explanation:

To find the coordinates of point P that divides line segment AB in the ratio 1:3, we'll use the section formula. Let's denote the coordinates of point P as (x, y). The formula for finding the coordinates of a point dividing a line segment with endpoints (x1, y1) and (x2, y2) in the ratio m:n is:

x = (mx2 + nx1)/(m+n)

y = (my2 + ny1)/(m+n)

Given A(2,4) and B(10,12), we need to find point P in the ratio 1:3. Plugging the values into the formula:

x =(1*10 + 3*2)/(1+3) = 10 + 6/4 = 16/4 = 4

y =(1*12 + 3*4)/(1+3) =12 + 12/4 = 24/4 = 6

Hence, the coordinates of point P that partitions AB in the ratio 1:3 are (4,6). This result aligns with the choice b) (4,6).

This calculation utilizes the section formula, which finds the coordinates of a point dividing a line segment into a given ratio. By substituting the coordinates of points A and B into the formula, we determine the values of x and y for point P. This method ensures accurate positioning of P along the line segment AB based on the specified ratio, enabling precise location determination in the coordinate plane. Therefore the correct answer is option b.