What are the coordinates of a point A on the line joining M(-19,-7) and N(1,8) if MA:AN = 2:3? help please?

Question

What are the coordinates of a point A on the line joining M(-19,-7) and N(1,8) if MA:AN = 2:3?

will someone help me out with this? an assignment in math of my kid. thanks

Answer 1

Find the equation of the line through the two points:

y - y₁ = (y₁ - y₂)(x - x₁) / (x₁ - x₂)

y - (-7) = (-7 - 8)(x + 19) / (-19 - 1)

y = -15(x + 19) / (-20) - 7

y = (3/4)x + 29/4

Let A be (x₀, y₀). Because the ratio of MA:AN also applies to the x- and y-coordinates, we can write:

(x₀ - (-19)) / (1 - x₀) = 2/3

3(x₀ + 19) = 2(1 - x₀)

5x₀ = -55

x₀ = -11

y₀ = (3/4)x₀ + 29/4 = -33/r + 29/4 = -1

So A is (-11, -1).

You can double check this by using the distance formula. You'll find that MN = 25, MA = 10 and AN = 15. Thus, the ratio is correct: MA/AN = 10/15 = 2/3.