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Sagot :
To find the coordinates of a point that is a given fraction of the way from one point to another, you can use the section formula for dividing a line in a given ratio. Here, we are asked for the coordinates of a 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].
Here's the step-by-step solution:
1. Identify the coordinates of the given points:
- Point [tex]\(A\)[/tex] has coordinates [tex]\((-8, -9)\)[/tex].
- Point [tex]\(B\)[/tex] has coordinates [tex]\((24, -1)\)[/tex].
2. Identify the fraction of the way from [tex]\(A\)[/tex] to [tex]\(B\)[/tex]:
- The fraction is [tex]\(\frac{3}{8}\)[/tex].
3. Determine the formula for the coordinates of the point [tex]\(P\)[/tex] that divides the line segment [tex]\(AB\)[/tex] in the ratio [tex]\(k:m\)[/tex]:
- If [tex]\(P\)[/tex] divides the line segment [tex]\(AB\)[/tex] in the ratio [tex]\(k:m\)[/tex], the coordinates of [tex]\(P(x, y)\)[/tex] are given by:
[tex]\[ P_x = \frac{m \cdot A_x + k \cdot B_x}{m + k} \][/tex]
[tex]\[ P_y = \frac{m \cdot A_y + k \cdot B_y}{m + k} \][/tex]
4. Apply the ratio for [tex]\(\frac{3}{8}\)[/tex]:
- Here, [tex]\(k = 3\)[/tex] and [tex]\(m = 8 - 3 = 5\)[/tex] (since [tex]\(\frac{3}{8}\)[/tex] of the way corresponds to a ratio of [tex]\(3\)[/tex] parts to [tex]\(5\)[/tex] parts).
5. Calculate the x-coordinate of the point:
[tex]\[ P_x = \frac{5 \cdot (-8) + 3 \cdot 24}{5 + 3} \][/tex]
[tex]\[ P_x = \frac{5 \cdot (-8) + 3 \cdot 24}{8} \][/tex]
[tex]\[ P_x = \frac{-40 + 72}{8} \][/tex]
[tex]\[ P_x = \frac{32}{8} \][/tex]
[tex]\[ P_x = 4 \][/tex]
6. Calculate the y-coordinate of the point:
[tex]\[ P_y = \frac{5 \cdot (-9) + 3 \cdot (-1)}{5 + 3} \][/tex]
[tex]\[ P_y = \frac{5 \cdot (-9) + 3 \cdot (-1)}{8} \][/tex]
[tex]\[ P_y = \frac{-45 + (-3)}{8} \][/tex]
[tex]\[ P_y = \frac{-48}{8} \][/tex]
[tex]\[ P_y = -6 \][/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 answer is [tex]\( \boxed{(4,-6)} \)[/tex].
Here's the step-by-step solution:
1. Identify the coordinates of the given points:
- Point [tex]\(A\)[/tex] has coordinates [tex]\((-8, -9)\)[/tex].
- Point [tex]\(B\)[/tex] has coordinates [tex]\((24, -1)\)[/tex].
2. Identify the fraction of the way from [tex]\(A\)[/tex] to [tex]\(B\)[/tex]:
- The fraction is [tex]\(\frac{3}{8}\)[/tex].
3. Determine the formula for the coordinates of the point [tex]\(P\)[/tex] that divides the line segment [tex]\(AB\)[/tex] in the ratio [tex]\(k:m\)[/tex]:
- If [tex]\(P\)[/tex] divides the line segment [tex]\(AB\)[/tex] in the ratio [tex]\(k:m\)[/tex], the coordinates of [tex]\(P(x, y)\)[/tex] are given by:
[tex]\[ P_x = \frac{m \cdot A_x + k \cdot B_x}{m + k} \][/tex]
[tex]\[ P_y = \frac{m \cdot A_y + k \cdot B_y}{m + k} \][/tex]
4. Apply the ratio for [tex]\(\frac{3}{8}\)[/tex]:
- Here, [tex]\(k = 3\)[/tex] and [tex]\(m = 8 - 3 = 5\)[/tex] (since [tex]\(\frac{3}{8}\)[/tex] of the way corresponds to a ratio of [tex]\(3\)[/tex] parts to [tex]\(5\)[/tex] parts).
5. Calculate the x-coordinate of the point:
[tex]\[ P_x = \frac{5 \cdot (-8) + 3 \cdot 24}{5 + 3} \][/tex]
[tex]\[ P_x = \frac{5 \cdot (-8) + 3 \cdot 24}{8} \][/tex]
[tex]\[ P_x = \frac{-40 + 72}{8} \][/tex]
[tex]\[ P_x = \frac{32}{8} \][/tex]
[tex]\[ P_x = 4 \][/tex]
6. Calculate the y-coordinate of the point:
[tex]\[ P_y = \frac{5 \cdot (-9) + 3 \cdot (-1)}{5 + 3} \][/tex]
[tex]\[ P_y = \frac{5 \cdot (-9) + 3 \cdot (-1)}{8} \][/tex]
[tex]\[ P_y = \frac{-45 + (-3)}{8} \][/tex]
[tex]\[ P_y = \frac{-48}{8} \][/tex]
[tex]\[ P_y = -6 \][/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 answer is [tex]\( \boxed{(4,-6)} \)[/tex].
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