Question

What are the coordinates of the point on the directed line segment from (-10, 2) to

(-3, -5) that partitions the segment into a ratio of 4 to 3?

Answer 1

(-6, -2)

Explanation:

For points A(-10, 2) and B(-3, -5), you want the point P that makes AP/PB = 4/3.

Setup

Using (x, y) for the coordinates of P, we have ...

AP/PB = 4/3

((x, y) -(-10, 2))/((-3, -5) -(x, y)) = 4/3

Solution

This simplifies to ...

(x+10, y-2)/(-3-x, -5-y) = 4/3

Cross multiplying gives ...

3(x +10, y -2) = 4(-3 -x, -5 -y)

(3x+30, 3y-6) = (-12-4x, -20-4y)

Treating these equations separately, we have ...

3x +30 = -12 -4x ⇒ 7x = -42 ⇒ x = -6

3y -6 = -20 -4y ⇒ 7y = -14 ⇒ y = -2

The point that partitions the segment is (-6, -2).

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Additional comment

The point that partitions AB in the ratio m/n is ...

P = (mB +nA)/(m+n)

P = (4(-3, -5) +3(-10, 2))/(4+3) = (-12-30, -20+6)/7 = (-42, -14)/7 = (-6, -2)

Above, we started from the basic requirement, rather than using the formula that results from that requirement.