Given:
Coordinates of point A: (-2, -2)
Coordinates of point B: (2, -4)

We need to find the coordinates of point P such that AP = (3/7)AB and P lies on the line segment AB.

Approach:
To find the coordinates of point P, we can use the concept of section formula. The section formula states that the coordinates of a point dividing a line segment AB in the ratio m:n (where m + n = 1) are given by:

P(x, y) = ((n * Ax + m * Bx)/(m + n), (n * Ay + m * By)/(m + n))

In our case, we know that AP = (3/7)AB. So, m = 3 and n = 7-3 = 4.

Calculations:
Let's substitute the values into the section formula to find the coordinates of point P.

P(x, y) = ((4 * Ax + 3 * Bx)/(3 + 4), (4 * Ay + 3 * By)/(3 + 4))

Substituting the coordinates of point A and B, we get:

P(x, y) = ((4 * (-2) + 3 * 2)/(3 + 4), (4 * (-2) + 3 * (-4))/(3 + 4))

P(x, y) = ((-8 + 6)/7, (-8 - 12)/7)

P(x, y) = (-2/7, -20/7)

Therefore, the coordinates of point P are (-2/7, -20/7).

Explanation:
- We started by using the section formula, which helps us find the coordinates of a point dividing a line segment in a given ratio.
- By substituting the values of point A, point B, and the given ratio, we obtained the coordinates of point P.
- The section formula is derived from the concept of similar triangles, where the ratios of corresponding sides are equal.
- In this case, we divided the line segment AB into two parts, AP and PB, such that AP is 3/7 times the length of AB.
- The resulting coordinates of point P are (-2/7, -20/7), which indicates its position on the line segment AB.