Find the point on a segment that is 3/8 of the way from A(1,2) to B(8,7).


To find the point that is \( \frac{3}{8} \) of the way from point \( A(1, 2) \) to point \( B(8, 7) \), we can use the formula for finding a point that divides a segment into a certain ratio.

The formula for finding a point that divides a line segment between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \) is:

\[
P = \left( \frac{n \cdot x_1 + m \cdot x_2}{m+n}, \frac{n \cdot y_1 + m \cdot y_2}{m+n} \right)
\]

Here, \( m = 3 \) and \( n = 8 - 3 = 5 \), since we are looking for the point that is \( \frac{3}{8} \) of the way.

Substituting the values of \( A(1, 2) \) and \( B(8, 7) \):

\[
P = \left( \frac{5 \cdot 1 + 3 \cdot 8}{5+3}, \frac{5 \cdot 2 + 3 \cdot 7}{5+3} \right)
\]

Now, calculating each coordinate:

\[
P_x = \frac{5 \cdot 1 + 3 \cdot 8}{8} = \frac{5 + 24}{8} = \frac{29}{8} = 3.625
\]

\[
P_y = \frac{5 \cdot 2 + 3 \cdot 7}{8} = \frac{10 + 21}{8} = \frac{31}{8} = 3.875
\]

Thus, the point that is \( \frac{3}{8} \) of the way from \( A(1, 2) \) to \( B(8, 7) \) is \( P(3.625, 3.875) \).

Would you like further details or have any questions?

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Here are 5 related questions:
1. How do you find the midpoint of a line segment between two points?
2. What is the general formula for dividing a segment in any given ratio?
3. How would you find the point that is \( \frac{1}{3} \) of the way from one point to another?
4. Can you calculate the length of the segment between \( A(1,2) \) and \( B(8,7) \)?
5. What happens if the ratio is negative, i.e., if the point is outside the segment?

**Tip:** You can use the same formula for dividing a segment in any ratio, not just between 0 and 1!

Find the point on a segment that is 3/8 of the way from A(1,2) to B(8,7).

Discover how to find the point that is 3/8 of the way from point A(1,2) to point B(8,7) using the point division formula. This solution covers key geometry concepts and the ratio division of a line segment, suitable for students in Grades 8-10.

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