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What are the coordinates of the point that is [tex]\frac{3}{8}[/tex] of the way from [tex]A(-8, -9)[/tex] to [tex]B(24, -1)[/tex]?

A. [tex](-6, 4)[/tex]
B. [tex](4, -6)[/tex]
C. [tex](-2, 4)[/tex]
D. [tex](12, -4)[/tex]

Sagot :

To determine the coordinates of the point that is [tex]\(\frac{3}{8}\)[/tex] of the way from point [tex]\(A(-8, -9)\)[/tex] to point [tex]\(B(24, -1)\)[/tex], we can use the section formula for dividing a line segment in a given ratio.

The section formula states that the coordinates of a point dividing the line segment joining [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\(m:n\)[/tex] are given by:

[tex]\[ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \][/tex]

Here, [tex]\(A(x_1, y_1) = (-8, -9)\)[/tex] and [tex]\(B(x_2, y_2) = (24, -1)\)[/tex], and the ratio [tex]\(m:n = \frac{3}{8}\)[/tex] implies [tex]\(m = 3\)[/tex] and [tex]\(n = 8 - 3 = 5\)[/tex].

Let's find the x-coordinate of the new point.

[tex]\[ x_{new} = \frac{m \cdot x_2 + n \cdot x_1}{m + n} = \frac{3 \cdot 24 + 5 \cdot (-8)}{3+5} = \frac{72 - 40}{8} = \frac{32}{8} = 4 \][/tex]

Next, let's find the y-coordinate of the new point.

[tex]\[ y_{new} = \frac{m \cdot y_2 + n \cdot y_1}{m + n} = \frac{3 \cdot (-1) + 5 \cdot (-9)}{3+5} = \frac{-3 - 45}{8} = \frac{-48}{8} = -6 \][/tex]

Therefore, the coordinates of the point that is [tex]\(\frac{3}{8}\)[/tex] of the way from [tex]\(A(-8, -9)\)[/tex] to [tex]\(B(24, -1)\)[/tex] are [tex]\((4, -6)\)[/tex].

So, the correct answer is:

B. [tex]\( (4, -6) \)[/tex]