Section Formula: Derivation, Types, Midpoint Formula

The ratio in which the point divides the line segment as well as the coordinates of the point can be found by using a simple formula in coordinate geometry called the Section Formula. A point on a line segment divides it into two separate parts, either equally or not. The main applications of the section formula in 2D geometry are to find the ratio in which the point divides the line segments as well as find the coordinates of the point that divides the line segments. Section formula is used to find the coordinates of a point that divides the line segment equally or in some ratio.

Key Terms: Section Formula, Midpoint Formula, Derivation, Internal Section, External Section

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What is Section Formula?

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In coordinate geometry, the Section formula is defined as the formula used for calculating the coordinates of the point which divides a line, either internally or externally, into two parts in the ratio m:n, where m and n are the ratios of the length of each separate line segment.

Let us take two points P and Q, with their coordinates (x₁, y₁) and (x₂, y₂) respectively, and PQ is the line segment that joins these two points. R is the point that divides the line segment PQ internally in the ratio m and n. 

The coordinates of point R i.e. (x, y), can be calculated by using the section formula as follows:

R(x, y) = \((\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n})\)

Section formula is mainly divided into two types:

1. Internal Section Formula
2. External Section Formula

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Section Formula Derivation

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Section formula is derived by using two similar right angles in a geometric plane. The hypotenuse of these right angles is given in the ratio m: n. The construction for the derivation of the section formula is as follows:

Section Formula Derivation

In the two right triangles AQP and BRP,

∠PAQ = ∠BPR (corresponding angles)

∠PQA = ∠BRP (90° angles)

By AA similarity criteria,

\(\frac{AQ}{PR}=\frac{PQ}{BR}=\frac{AP}{BP}= \frac{m}{n}\) -(1)

Also,

AQ = x – x₁ -(2)

PR = x₂ – x -(3) 

From (1), (2) & (3), we have,

\(\frac{x – x_1}{x_2 – x} = \frac{m}{n}\)

On solving for x, we get:

x= \(\frac{mx_2+nx_1}{m+n}\)-(A)

Similarly for y, we have,

PQ = y – y₁ -(4)

BR = y₂ – y -(5) 

From (1), (4) & (5), we have,

\(\frac{y – y_1}{y_2 – y}=\frac{m}{n}\)

On solving for y, we get:

y= \(\frac{my_2+ny_1}{m+n}\) -(B)

Hence, from equations (A) and (B),

P(x, y) =\( (\frac{mx_2+nx_1}{m+n} , \frac{my_2+ny_1}{m+n})\)

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Types Of Section Formula

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The two classifications of section formulas are: Internal Section Formula and External Section Formula.

Internal Section Formula

The internal section formula, also known as the section formula for internal division, is a formula used for finding the coordinates of a point that lies internally or within the original line segment.

Here, the coordinates of point R will be:

R (x, y) = \( (\frac{mx_2+nx_1}{m+n} , \frac{my_2+ny_1}{m+n})\)

The internal section formula is the same as the section formula.

External Section Formula

The external section formula, also known as the section formula for external division, is a formula used for finding the coordinates of a point dividing a line segment into two parts, which lie externally or outside the original line segment.

Here, the coordinates of point C will be:

C (x, y) = \( (\frac{mx_2-nx_1}{m-n} , \frac{my_2-ny_1}{m-n})\)

Derivation of External Section Formula

The derivation of the external section formula is similar to the internal section formula in the sense that the same graph construction is used and the same principles are being applied. The construction for the derivation of the external section formula is as follows:

In the above diagram,

AM = PR = OR – OP = x – x₁ -(1)

BN = QR = OR – OQ = x – x₂ -(2)

Similarly,

CM = RC – MR = y – y₁ -(3)

CN = RC – RN = y – y₂ -(4)

In the two right triangles AMC and BNC,

∠ACM = ∠BCM (same angle)

∠AMC = ∠BNC (90° angles)

By AA similarity criteria,

\(\frac{AM}{BN} = \frac{CM}{CN} = \frac{AC}{BC} = \frac{m}{n}\) -(5)

Substituting the values in (5) for (1),(2),(3) & (4),

\(\frac{m}{n} = \frac{x - x_1}{ x- x_2} = \frac{y- y_1}{y- y_2}\)

By solving for x, we get:

m (x – x₂) = n (x – x₁)

(m – n) x = (mx₂ – nx₁)

x = \((\frac{mx_2 – nx_1}{m – n})\) -(A)

By solving for y, we get:

m(y – y₂) = n (y – y₁)

(m – n) y = (my₂ – ny₁)

y = \((\frac{my_2-ny_1}{m-n}) \)-(B)

Hence, from equations (A) and (B),

C(x, y) = \( (\frac{mx_2-nx_1}{m-n} , \frac{my_2-ny_1}{m-n})\)

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Mid-point Formula

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The mid-point formula is used to find the coordinates of a point that lies exactly in the middle of a line segment, dividing the line segment into two halves. Since the length of each half is equal, the ratio between them is 1:1.

Since point C lies inside AB, we can use the internal section formula:

C (x, y) = \( (\frac{mx_2+nx_1}{m+n} , \frac{my_2+ny_1}{m+n})\)

Since the m and n are equal, substituting m, n for 1 gives:

C (x, y) = \((\frac{x_2+x_1}{1+1}, \frac{y_2+y_1}{1+1})\)

Hence, the mid-point formula is:C (x, y) = \((\frac{x_2+x_1}{2}, \frac{y_2+y_1}{2})\)

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Things to Remember

• Section formula is used to find coordinates of a point which divides the line segment into two parts.
• Section formula is only applicable to line segments.
• The ratio in which the point divides the line segment is denoted by m: n.
• A line can be divided both internally and externally.
• Internal section formula is \( (\frac{mx_2+nx_1}{m+n} , \frac{my_2+ny_1}{m+n})\), while external section formula is \( (\frac{mx_2-nx_1}{m-n} , \frac{my_2-ny_1}{m-n})\).
• Mid-point formula is used to find the coordinates of a point that exactly divides the line segment into two halves.

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