Question

Find a point P on the segment with endpoints M(-2,2) and N(4, -1) that partitions the segment, starting at point N, two-thirds of the way to point M. The coordinates for P are _____.

Answer 1

The coordinates of point P that partitions the segment with endpoints M(-2,2) and N(4, -1) two-thirds of the way from N to M are (0, 1). The point is determined using the section formula, where the ratio is 2:1 favoring point M.

Step-by-step explanation:

To find a point P on the segment with endpoints M(-2,2) and N(4, -1) that partitions the segment two-thirds of the way from N to M, you can use the formula for dividing a segment into a given ratio, which is a form of section formula. In this case, the ratio is 2:1 (since we want two-thirds of the way from N to M) and we'll denote this as a ratio of λ:1, where λ = 2.

The coordinates of P can be found using the following formulas:

• xP = (1*λ*xM + xN) / (λ + 1)
• yP = (1*λ*yM + yN) / (λ + 1)

Plugging the values in, we get:

• xP = (2*-2 + 4) / (2 + 1) = (2*-2 + 4) / 3 = (-4 + 4) / 3 = 0 / 3 = 0
• yP = (2*2 - 1) / (2 + 1) = (4 - 1) / 3 = 3 / 3 = 1

Therefore, the coordinates of P are (0, 1).