Section Formula and Mid-point

Chapter in a nutshell: The section formula gives the coordinates of a point that divides a line segment in a given ratio $m_1:m_2$; the mid-point is the special case of ratio $1:1$. Related results include the centroid of a triangle and (prerequisite) the distance formula.

1. Key Ideas

• A point P divides the segment joining $A(x_1,y_1)$ and $B(x_2,y_2)$ in the ratio $m_1:m_2$ (internally).
• Mid-point = the point dividing in $1:1$.
• Centroid = the point where a triangle's medians meet (divides each median 2:1).

2. Formulas (quick reference — all of them)

• Section formula (internal division, ratio $m_1:m_2$):

• Mid-point (ratio 1:1):

• Centroid of a triangle $A,B,C$:

• Finding the ratio in which a point P divides AB: take $m_1:m_2=\dfrac{x_P-x_1}{x_2-x_P}$ (or use y-coordinates).
• (Prerequisite) Distance formula: $AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.
• (Prerequisite) Distance from origin: $\sqrt{x^2+y^2}$.

3. Prerequisite Ideas

• Cartesian plane / coordinates; ratio (from the Ratio chapter).

4. Worked Example (one, for the method)

5. Common Mistakes to Avoid

• Putting $m_1$ with $x_1$ — note the cross pattern: $m_1$ multiplies $x_2$ (the far point).
• Using the section formula when only the mid-point is needed (or vice versa).
• Sign errors with negative coordinates.
• Forgetting the centroid uses ÷3 (sum of all three).

6. Likely Exam Questions (with crisp answers)

1. State the section formula. → $\left(\frac{m_1x_2+m_2x_1}{m_1+m_2},\frac{m_1y_2+m_2y_1}{m_1+m_2}\right)$.
2. State the mid-point formula. → $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.
3. State the centroid formula. → $\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)$.
4. Mid-point of (2,4) and (6,8)? → (4, 6).
5. In what ratio does the mid-point divide a segment? → 1:1.
6. Recall the distance formula. → $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.
7. In what ratio do medians meet at the centroid? → 2:1.
8. Centroid of (0,0), (6,0), (0,9)? → (2, 3).
9. How do you find the ratio in which P divides AB? → $\frac{x_P-x_1}{x_2-x_P}$.
10. Distance of (3,4) from origin? → 5.

Extended & Prerequisite Formulas (full)

• Section: $\left(\tfrac{m_1x_2+m_2x_1}{m_1+m_2},\tfrac{m_1y_2+m_2y_1}{m_1+m_2}\right)$; Mid-point $\left(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\right)$; Centroid $\left(\tfrac{\sum x}{3},\tfrac{\sum y}{3}\right)$.
• Ratio P divides AB = (x_P−x₁):(x₂−x_P).
• Prereq: Distance $=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$; collinear if area of triangle = 0; area $=\tfrac12|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|$.