find the coordinates of the point 3/4 of the way from a to b

Question

the coordinates of b is (-3,3) the coordinates of a is (-5,-4)

Answer 1

Segment Ratio Formula: The coordinates of the point that splits a segment into an arbitrary ratio, n:m, or fractional part, n ⁄ (n+m) , are given by:

x = [n ⁄ (n+m)] (x₂ - x₁) + x₁ and y = [n ⁄ (n+m)] (y₂ - y₁) + y₁

Where [n ⁄ (n+m)] is the desired distance ratio from the starting point, both n and m are positive integers, and ( x₁, y₁) are the coordinates of one end point and ( x₂, y₂)  are the coordinates of the other endpoint.

For this problem, n = 3 and m = 1 since (3 ⁄ (3+1) = 3 ⁄ 4, ( x₁, y₁) = (-5, -4) and ( x₂, y₂) = (-3,3)

x = [n ⁄ (n+m)] (x₂ - x₁) + x₁ = {(3 ⁄ 4)*[ (-3 )- (-5) ]} + (-5) = [(3/4)*(2)] - 5 = [3/2] - 10/3 = -7 ⁄ 2

y = [n ⁄ (n+m)] (y₂ - y₁) + y₁ ={ (3/4)*[3 - (- 4)] } + (-5) = [(3/4) (7) ] - 5 = [21/4] - [20/4] = 1 ⁄ 4

The coordinates of the point are ( -7/2 , 1/4)