find 1/3 of the way through the points (5,1) and (-1,7)

To find a point that is \( \frac{1}{3} \) of the way from point \( A(5, 1) \) to point \( B(-1, 7) \), we can use the section formula. The section formula for a point \( P(x, y) \) dividing the line segment joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \) is:

\[
P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

Here, \( m = 1 \) and \( n = 2 \), because we want the point to be \( \frac{1}{3} \) of the way from \( A \) to \( B \).

Given points \( A(5, 1) \) and \( B(-1, 7) \):

\[
P(x, y) = \left( \frac{1(-1) + 2(5)}{1+2}, \frac{1(7) + 2(1)}{1+2} \right)
\]

Now, calculate the coordinates:

\[
x = \frac{-1 + 10}{3} = \frac{9}{3} = 3
\]

\[
y = \frac{7 + 2}{3} = \frac{9}{3} = 3
\]

So, the point \( \frac{1}{3} \) of the way from \( (5, 1) \) to \( (-1, 7) \) is \( (3, 3) \).

Would you like further details or have any questions?

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Here are five related questions:
1. How do you find the midpoint between two points?
2. What is the formula for the distance between two points?
3. How do you find the ratio in which a line segment is divided by a point?
4. How do you generalize the section formula for three dimensions?
5. How can you verify if three points are collinear?

**Tip:** When using the section formula, always ensure the correct order of points to get the desired direction for the division.

Learn how to find a point that is 1/3 of the way from point (5, 1) to (-1, 7) using the section formula in coordinate geometry. This problem involves calculating coordinates and applying mathematical ratios in a line segment division.

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