To find the coordinates of point P that divides the line segment AB in the ratio [tex]\(3:1\)[/tex], given the coordinates of points [tex]\(A = (0, 7)\)[/tex] and [tex]\(B = (-4, 2)\)[/tex], we use the section formula.
The section formula states that if a point [tex]\(P\)[/tex] divides the line segment joining points [tex]\(A(x_1, y_1)\)[/tex] and [tex]\(B(x_2, y_2)\)[/tex] in the ratio [tex]\(m:n\)[/tex], the coordinates of [tex]\(P\)[/tex] are:
[tex]\[ P(x, y) = \left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right) \][/tex]
Here, [tex]\(A(x_1, y_1) = (0, 7)\)[/tex] and [tex]\(B(x_2, y_2) = (-4, 2)\)[/tex]. The given ratio is [tex]\(m:n = 3:1\)[/tex].
Let's substitute these values into the section formula.
For the [tex]\(x\)[/tex]-coordinate of [tex]\(P\)[/tex]:
[tex]\[ x = \frac{3(-4) + 1(0)}{3 + 1} \][/tex]
[tex]\[ x = \frac{-12 + 0}{4} \][/tex]
[tex]\[ x = \frac{-12}{4} \][/tex]
[tex]\[ x = -3 \][/tex]
For the [tex]\(y\)[/tex]-coordinate of [tex]\(P\)[/tex]:
[tex]\[ y = \frac{3(2) + 1(7)}{3 + 1} \][/tex]
[tex]\[ y = \frac{6 + 7}{4} \][/tex]
[tex]\[ y = \frac{13}{4} \][/tex]
[tex]\[ y = 3.25 \][/tex]
Therefore, the coordinates of point [tex]\(P\)[/tex] are [tex]\((-3, 3.25)\)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{(-3, 3.25)} \][/tex]
