Exercise 7.2 Question 1 – Section Formula
Find the coordinates of the point which divides the join of \((1, -2)\) and \((3, 4)\) in the ratio \(2:3\) internally.
🖼️ Visual Representation
Figure: Point P divides line segment AB in the ratio 2:3 internally
📋 Given
- Point A: \((1, -2)\) — coordinates \((x_1, y_1)\)
- Point B: \((3, 4)\) — coordinates \((x_2, y_2)\)
- Ratio: m:n = 2:3 (internal division)
🎯 To Find
The coordinates of point P(x, y) that divides the line segment AB in the ratio 2:3 internally.
📐 Formula
Section Formula (Internal Division):
If point P divides the line segment joining \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in the ratio \(m:n\) internally, then:
💡 Remember: For internal division, both m and n are positive, and P lies between A and B.
📝 Solution
- \(x_1 = 1\), \(y_1 = -2\)
- \(x_2 = 3\), \(y_2 = 4\)
- \(m = 2\), \(n = 3\)
The required point P has coordinates:
\[ P = \left(\frac{9}{5}, \frac{2}{5}\right) \]Or in decimal form: \(P = (1.8, 0.4)\)
or P = (1.8, 0.4)
✓ Verification
Let’s verify that P divides AB in the ratio 2:3 by calculating distances:
| Distance | Calculation | Result |
|---|---|---|
| AP | \(\sqrt{(\frac{9}{5}-1)^2 + (\frac{2}{5}-(-2))^2}\) | \(\sqrt{(\frac{4}{5})^2 + (\frac{12}{5})^2} = \frac{4\sqrt{10}}{5}\) |
| PB | \(\sqrt{(3-\frac{9}{5})^2 + (4-\frac{2}{5})^2}\) | \(\sqrt{(\frac{6}{5})^2 + (\frac{18}{5})^2} = \frac{6\sqrt{10}}{5}\) |
| AP:PB | \(\frac{4\sqrt{10}/5}{6\sqrt{10}/5} = \frac{4}{6}\) | \(= \frac{2}{3}\) or 2:3 ✓ |
The ratio AP:PB = 2:3 is verified! ✓
💡 Understanding the Section Formula
Why does the section formula work?
The section formula is based on the concept of weighted average:
- When P divides AB in ratio m:n, it means AP:PB = m:n
- P is closer to A if m < n, and closer to B if m > n
- The formula gives more “weight” to the point that is farther away
- In our case, m=2 and n=3, so P is closer to A than to B
Special Cases:
- Midpoint (m=n=1): \(P = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\)
- Trisection (m:n = 1:2 or 2:1): Divides into three equal parts
- External Division: Uses the formula with a minus sign (covered in later questions)
⚠️ Common Mistakes
✅ Correct: Remember: m goes with the second point (B), n goes with the first point (A).
✅ Correct: Always be careful with negative coordinates.
✅ Correct: For internal division, denominator is always \(m + n\).
✅ Correct: If ratio is m:n, then AP:PB = m:n (not PB:AP).
💡 Key Points
- The section formula finds coordinates of a dividing point
- For internal division: \(P = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)\)
- The point P lies between A and B for internal division
- When m:n = 1:1, the formula gives the midpoint
- Always identify \(x_1, y_1, x_2, y_2, m, n\) before substituting
- The section formula is a weighted average of coordinates
- Verify your answer by checking if AP:PB = m:n
- Be careful with negative coordinates in calculations