find 2/3 of the way through the points (5,1) and (-1,7)

To find the point that is \( \frac{2}{3} \) of the way from point \( A(5, 1) \) to point \( B(-1, 7) \), we again 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, \( m = 2 \) and \( n = 1 \), because we want the point to be \( \frac{2}{3} \) of the way from \( A \) to \( B \).

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

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

Now, calculate the coordinates:

\[
x = \frac{-2 + 5}{3} = \frac{3}{3} = 1
\]

\[
y = \frac{14 + 1}{3} = \frac{15}{3} = 5
\]

So, the point \( \frac{2}{3} \) of the way from \( (5, 1) \) to \( (-1, 7) \) is \( (1, 5) \).

Would you like further details or have any questions?

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Here are five related questions:
1. How do you determine a point that is halfway between two points?
2. What is the formula for finding a point one-third of the way between two points?
3. How do you calculate the slope of a line between two points?
4. How does the section formula extend to finding points on a line segment in three dimensions?
5. Can you find the point that divides the segment in the ratio 3:2?

**Tip:** When calculating points on a line segment using the section formula, double-check the ratio to ensure it corresponds correctly to the required fraction of the way.

Learn how to find a point that is 2/3 of the way between the points (5, 1) and (-1, 7) using the section formula in coordinate geometry. This problem involves calculating the coordinates of a point dividing a line segment in a specific ratio. Suitable for students in Grades 9-10.

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