What is the difference between internal and external division?

Internal division: P lies between A and B, ratio m:n is positive. External division: P lies on the extension of AB, ratio m:n is negative (use absolute values for calculation; the sign determines whether P is on the outer or opposite extension).

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Section Ratio Calculator

Enter coordinates of two endpoints and division ratio to compute the division point

Point A

x\u2081

y\u2081

Point B

x\u2082

y\u2082

Ratio

m =

n =

(m : n)

Section Formula

P( (m\u00b7x\u2082 + n\u00b7x\u2081)/(m+n) , (m\u00b7y\u2082 + n\u00b7y\u2081)/(m+n) )

When m=n=1, this reduces to the midpoint formula P((x\u2081+x\u2082)/2, (y\u2081+y\u2082)/2). m\u00b7n < 0 for external division, m\u00b7n > 0 for internal division.

⚠ Note: n cannot be 0, otherwise the denominator is zero. For external division, m\u00b7n<0, meaning m and n have opposite signs.

What Is a Section Point?

A section point is an important concept in analytic geometry. Given points A(x\u2081,y\u2081) and B(x\u2082,y\u2082), if point P divides segment AB in ratio m:n (from A to B), then P is called the division point of AB.

Internal Division

When m>0 and n>0, P lies between A and B. For example, m:n=1:1 is the midpoint; m:n=1:2 means PA = (1/2) PB.

External Division

When m\u00b7n<0, P lies on the extension of AB. The formula still works, but |m+n| is smaller than |m|+|n|.

Midpoint Formula

When m=n=1, the section formula reduces to P((x\u2081+x\u2082)/2, (y\u2081+y\u2082)/2), the most common midpoint formula in coordinate geometry.

Vector Derivation

From vector relation AP/AB = m/(m+n): P(x) = x\u2081 + m/(m+n)\u00b7(x\u2082\u2212x\u2081) = (mx\u2082+nx\u2081)/(m+n), and similarly for y.

Teaching Example: Find the point dividing A(0,0), B(6,0) in ratio 1:2. P = ((1\u00d76+2\u00d70)/(1+2), (1\u00d70+2\u00d70)/(1+2)) = (2, 0). So AP:PB = 1:2, P is 1/3 of the total length from A.

Applications

Section Ratio Coordinate Calculation Internal & External Competition Math Exam Prep

Frequently Asked Questions

What is a section (division) point?▼

A division point divides a line segment in a specified ratio m:n. Internal: P on AB (m>0,n>0). External: P on extension (m\u00b7n<0). Midpoint: special case m=n=1.

How is the section formula derived?▼

From vector AP:PB = m:n. x = (mx\u2082+nx\u2081)/(m+n), y = (my\u2082+ny\u2081)/(m+n).

Internal vs external division?▼

Internal: P between A and B, m:n positive. External: P on extension, m:n negative. Sign determines which extension side.

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