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Lesson 3 of 4

Coordinate Geometry · Lesson 3 of 4

Section Formula

“If you’ve ever had to split a pizza bill down to the last slice based on who ate what, congratulations: you’ve already used the Section Formula in real life.”

Learning Objectives

• Understand what it means to divide a line segment in a ratio. • Learn the section formula for internal division. • Use the formula to find a point, a ratio, or the midpoint. • Apply the idea to trisection and simple geometric problems.

Introduction

Suppose two points are marked on a coordinate plane and joined by a line segment. Now imagine placing another point somewhere on that segment. We may want this new point to divide the segment equally, or in a given ratio such as (2:1), (3:2), or (4:3). How can we find its exact coordinates without measuring the diagram?

The section formula helps us find the coordinates of a point that divides a line segment in a given ratio. For example, if a point divides a segment in the ratio \(2:1\), the segment is split into three equal parts: two parts lie on one side of the point and one part lies on the other.

What Does 'Divides in a Ratio' Mean?

If a point P lies between A and B and AP : PB = m₁ : m₂, then P divides AB internally in the ratio m₁ : m₂. The word internally means that the point lies inside the segment, not outside it.

Cg L3 Internal Division
Internal Division of a Segment— Show A(x1,y1), B(x2,y2), and P(x,y) between them with AP:PB = m1:m2.

The Section Formula

By drawing horizontal and vertical helpers and using similar triangles, we can show that the coordinates of the dividing point are weighted averages of the end points.

Cg L3 Section Formula Derivation
How the Formula Is Built— Show A, B, P and helper perpendiculars to the axes. Indicate the similar-triangle idea in a simple visual manner.
Section FormulaLaTeX
How to Read the Formula

Multiply each x-coordinate and y-coordinate by the opposite ratio part, add them, and divide by the total ratio. This is why the formula is often called a weighted average.

Special Case: Midpoint

A midpoint divides a segment in the ratio 1 : 1. So, we get the midpoint formula directly from the section formula.

Midpoint FormulaLaTeX
Cg L3 Midpoint And Trisection
Midpoint and Trisection— Show a segment with midpoint and two trisection points to help students see common ratios.
A Useful Point to Remember

• If m₁ = m₂, then P is the midpoint of AB. • If m₁ < m₂, then P lies closer to A. • If m₁ > m₂, then P lies closer to B.

Worked Example: Find a Dividing Point

Problem
Find the coordinates of the point P that divides the segment joining A(2, -1) and B(10, 7) in the ratio 3 : 1 internally.

  1. 1.Here x₁ = 2, y₁ = -1, x₂ = 10, y₂ = 7, m₁ = 3 and m₂ = 1.
  2. 2.Use the section formula.
  3. 3.x = (3×10 + 1×2) / (3 + 1) = 32 / 4 = 8.
  4. 4.y = (3×7 + 1×(-1)) / (3 + 1) = 20 / 4 = 5.
  5. 5.Therefore, P = (8, 5).
Worked Example: Find the Ratio

Problem
The point P(-1, 4) divides the segment joining A(-5, 8) and B(7, -4) internally. Find the ratio AP : PB.

  1. 1.Let AP : PB = m₁ : m₂.
  2. 2.Using the x-coordinate: -1 = (m₁×7 + m₂×(-5)) / (m₁ + m₂).
  3. 3.So, -m₁ - m₂ = 7m₁ - 5m₂.
  4. 4.This gives 4m₂ = 8m₁, so m₂ = 2m₁.
  5. 5.Hence, m₁ : m₂ = 1 : 2.
  6. 6.Therefore, AP : PB = 1 : 2.
Worked Example: Find a Midpoint

Problem
Find the midpoint of the segment joining A(-3, 5) and B(9, -1).

  1. 1.Use the midpoint formula.
  2. 2.Midpoint = ((-3 + 9)/2, (5 + (-1))/2)
  3. 3.= (6/2, 4/2)
  4. 4.= (3, 2).
Worked Example: Find Points of Trisection

Problem
Find the points of trisection of the line segment joining A(1, 1) and B(10, 7).

  1. 1.The first trisection point divides AB in the ratio 1 : 2.
  2. 2.So, P = ((1×10 + 2×1)/3, (1×7 + 2×1)/3) = (4, 3).
  3. 3.The second trisection point divides AB in the ratio 2 : 1.
  4. 4.So, Q = ((2×10 + 1×1)/3, (2×7 + 1×1)/3) = (7, 5).
  5. 5.Therefore, the points of trisection are (4, 3) and (7, 5).
Worked Example: Division by an Axis

Problem
In what ratio does the x-axis divide the line segment joining A(2, 6) and B(-4, -3)? Also find the point of division.

  1. 1.Let the ratio be k : 1.
  2. 2.Using the section formula, the y-coordinate of the dividing point is (k×(-3) + 1×6)/(k + 1).
  3. 3.Because the point lies on the x-axis, its y-coordinate must be 0.
  4. 4.So, -3k + 6 = 0, which gives k = 2.
  5. 5.Hence, the ratio is 2 : 1.
  6. 6.Now find the x-coordinate: x = (2×(-4) + 1×2)/(2 + 1) = -6/3 = -2.
  7. 7.Therefore, the x-axis divides the segment in the ratio 2 : 1 at the point (-2, 0).
Common Mistakes

• Do not mix up m₁ and m₂. • Keep x-values together and y-values together. • For midpoint, remember the ratio is 1 : 1. • If the point lies on an axis, use the missing coordinate condition: x = 0 on y-axis, y = 0 on x-axis.

Quiz

Quick check

What ratio does the midpoint use?

Quick check

If a point lies on the x-axis, which coordinate is 0?

Quick check

The section formula gives the coordinates of a point that divides a line segment in a given ratio. This division is usually understood to be:

Practice Problems

Practice Problems
  1. Find the point dividing the segment joining (1, 4) and (9, 8) in the ratio 1 : 3 internally.
  2. Find the midpoint of the segment joining (-6, 7) and (4, -5).
  3. Find the points of trisection of the segment joining (0, 0) and (6, 9).

Key Takeaways

Key Takeaways

• The section formula finds the coordinates of a point that divides a line segment in a given ratio. • If P divides the segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m:n, then P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). • In the ratio m:n, m multiplies the coordinates of the second endpoint and n multiplies the coordinates of the first endpoint. • The midpoint formula is a special case of the section formula in which the dividing ratio is 1:1. • The section formula gives the exact coordinates of the dividing point, so no measurement from the diagram is required.

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Distance Formula

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Chapter Summary and Practice