Find the point P that partitions the line segement AB in the ratio of 2:1
Round to the hundredths place if necessary. Point a is (0,-3) Point B is (0,4)

To find the point \( P \) that partitions the line segment \( AB \) in the ratio \( 2:1 \), we use the section formula. The section formula for internal division is given by:

\[
P(x, y) = \left( \frac{m_2 \cdot x_1 + m_1 \cdot x_2}{m_1 + m_2}, \frac{m_2 \cdot y_1 + m_1 \cdot y_2}{m_1 + m_2} \right)
\]

Here:
- \( A(x_1, y_1) = (0, -3) \)
- \( B(x_2, y_2) = (0, 4) \)
- The ratio \( m_1 : m_2 = 2:1 \), so \( m_1 = 2 \) and \( m_2 = 1 \).

Now applying the section formula:

\[
P(x, y) = \left( \frac{1 \cdot 0 + 2 \cdot 0}{2+1}, \frac{1 \cdot (-3) + 2 \cdot 4}{2+1} \right)
\]

Simplifying each coordinate:

\[
P(x, y) = \left( 0, \frac{-3 + 8}{3} \right)
\]

\[
P(x, y) = \left( 0, \frac{5}{3} \right)
\]

\[
P(x, y) = \left( 0, 1.67 \right)
\]

Thus, the point \( P \) that partitions the line segment \( AB \) in the ratio \( 2:1 \) is \( P(0, 1.67) \).

Would you like further details or have any questions?

Here are some related questions:
1. How do you find the midpoint of a line segment?
2. How do you find a point that partitions a line segment externally?
3. How can the section formula be applied in three dimensions?
4. How do you determine the distance between two points on a coordinate plane?
5. What is the significance of a 1:1 partition ratio?

**Tip:** Always double-check the ratio being used in the section formula to ensure correct application!

Find the point P that partitions the line segment AB in the ratio of 2:1. Point A is (0,-3) and Point B is (0,4).

Learn how to find the point that partitions the line segment AB in the ratio 2:1 using the section formula. This detailed explanation involves applying the section theorem to points A(0, -3) and B(0, 4). The solution covers coordinate geometry and ratios, making it suitable for students in Grades 9-10.

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