Final answer:
To find the coordinates of point P dividing segment AK with endpoints A (1,2) and K (8,12) in a 3:2 ratio, use the section formula to get P's coordinates as (5.2, 8).
Step-by-step explanation:
To find the coordinates of point P that divides the segment AK with endpoints A (1,2) and K (8,12) into a ratio of 3:2, we can use the section formula. This is a mathematical method used in coordinate geometry to find a point that divides a line segment into a given ratio.
The section formula for point P(x, y) that divides the line segment connecting A(x1, y1) and K(x2, y2) is given by:
x = (m*x2 + n*x1) / (m + n)
y = (m*y2 + n*y1) / (m + n)
Where m and n are the ratios in which point P divides the segment, with m being the portion from A to P and n being the portion from P to K.
Substituting the given values:
- Let m = 3 and n = 2, the given ratio.
- A's coordinates x1 = 1 and y1 = 2.
- K's coordinates x2 = 8 and y2 = 12.
Calculating x coordinate:
x = (3*8 + 2*1) / (3 + 2) = (24 + 2) / 5 = 26 / 5 = 5.2
Calculating y coordinate:
y = (3*12 + 2*2) / (3 + 2) = (36 + 4) / 5 = 40 / 5 = 8
Therefore, the coordinates of point P are (5.2, 8).