Section formula — internal and external division: Foundations

Before you can read the parent note, you need to own every symbol it throws at you. Below, each one is built from nothing: plain words → the picture → why the topic can't live without it. Read top to bottom; each rung stands on the one below it.

1. A point and its address $(x, y)$

Figure s01 — An address $(x, y)$ : walk right, then up. The burnt-orange arrow is the horizontal move, the teal arrow the vertical move; the plum dot is where you land.

Figure — Section formula — internal and external division

The first number $x$ is the horizontal position. Look at the burnt-orange arrow: it slides you sideways.
The second number $y$ is the vertical position — the teal arrow lifts you up.
Order matters: $(3, 1)$ and $(1, 3)$ are different spots. That is why we say ordered pair.

Why the topic needs it: the whole game is to find the address of a splitting point $P$ . No addresses, nothing to compute.

2. Subscripts: $x_{1}, y_{1}, x_{2}, y_{2}$

So when we have two points we write: $A = (x_{1}, y_{1}), B = (x_{2}, y_{2}) .$

$x_{1}$ reads "x-one" = the horizontal address of point $A$ .
$y_{2}$ reads "y-two" = the vertical address of point $B$ .

The splitting point itself gets the plain letters: we write $P = (x, y)$ , so that $x$ means "the horizontal coordinate of $P$ " and $y$ means "the vertical coordinate of $P$ ". These two unknowns are exactly what the whole formula is hunting for.

Why the topic needs it: we juggle three points at once ( $A$ , $B$ , and the unknown $P$ ). Without labels we couldn't tell them apart.

3. A line segment $A B$ and its length $A P$ , $P B$

Figure s02 — Segment $A B$ split by $P$ into pieces $A P$ and $P B$ . Teal dot is $A$ , orange dot is $B$ , plum dot is the splitting point $P$ sitting between them.

$A P$ = how long the rope is from $A$ to the middle spot $P$ .
$P B$ = how long from $P$ onward to $B$ .
$A P + P B = A B$ only when $P$ sits between $A$ and $B$ (that is internal division — introduced in §7).

Why the topic needs it: the splitting is described entirely by comparing these two lengths.

4. Directed (signed) segments $\overline{A P}$

Figure s05 — Directed lengths carry a sign. With "left → right" chosen as positive, the forward hop $\overline{A P}$ is positive; a hop that doubles back is negative.

Ordinary length $A P$ is always $\geq 0$ (it's a distance).
Directed length $\overline{A P}$ can be negative — the minus sign is a memo that says "I had to walk backward".
Reversing the letters flips the sign: $\overline{P A} = - \overline{A P}$ .

Why the topic needs it: external division is exactly the case where one directed length is negative. Without signed segments, the parent note's " $n \to - n$ " trick would be magic; with them, it's just bookkeeping.

5. Ratio $m : n$ — comparing two lengths

We usually write a ratio as a fraction to compute with it: $\frac{A P}{P B} = \frac{m}{n} .$

$m$ = the number of parts in the first piece $A P$ (the piece touching $A$ ).
$n$ = the number of parts in the second piece $P B$ (the piece touching $B$ ).

Why the topic needs it: " $P$ divides $A B$ in ratio $m : n$ " IS the input to the section formula. Everything else is the machine that eats this ratio and spits out coordinates.

6. Between vs. beyond — internal and external

Figure s03 — Between (internal) vs beyond (external). Top row: $P$ lies between $A$ and $B$ , both plum arrows point the same way. Bottom row: $P$ is beyond $B$ , so the second hop turns around and one sign flips.

Internal (top row): both directed hops $\overline{A P}$ and $\overline{P B}$ point the same way, so both are positive. Lengths add: $A P + P B = A B$ .
External (bottom row): to go $A \to P$ then $P \to B$ you must turn around. Using §4, one directed length ( $\overline{P B}$ here) is negative.

Why the topic needs it: the parent gives two formulas, internal and external. They differ only by this direction/sign idea.

7. Slope, similar triangles, and the shadow ratio

Figure s06 — Same slope near $A$ and near $B$ . The tilt is measured by "rise over run"; on a straight line it is identical at both ends, which is what makes the two small triangles below scaled copies.

Figure s04 — Why the $x$ -shadows share the ratio. Dropping verticals from $A$ , $P$ , $B$ makes two similar triangles; the orange base is $x - x_{1}$ , the teal base is $x_{2} - x$ , and they split in the same ratio $m : n$ as the rope.

Now the engine. Drop straight-down verticals from $A$ , $P$ , and $B$ to the horizontal axis. Because the line's slope is the same near $A$ and near $B$ , the little triangle under $A P$ and the little triangle under $P B$ have equal angles — so they are similar. Matching sides therefore share one ratio, and the horizontal bases split exactly like the rope pieces: $\frac{x - x _{1}}{x _{2} - x} = \frac{A P}{P B} = \frac{m}{n} .$

$x - x_{1}$ = the burnt-orange horizontal "shadow" of piece $A P$ .
$x_{2} - x$ = the teal horizontal shadow of piece $P B$ .

This equation feeds the algebra below. Because the same similar-triangle argument works on the vertical shadows, an identical derivation gives $y$ .

8. The fraction bar and cross-multiplication

Here is the full clearing-and-solving walkthrough, so nothing is skipped. Start from the shadow-ratio just built in §7: $\frac{x - x _{1}}{x _{2} - x} = \frac{m}{n} .$ Step 1 — cross-multiply (kill the bars): $n (x - x_{1}) = m (x_{2} - x) .$ Step 2 — expand both sides: $n x - n x_{1} = m x_{2} - m x .$ Step 3 — gather every $x$ on the left, everything else on the right: $n x + m x = m x_{2} + n x_{1} .$ Step 4 — factor $x$ out of the left side: $(m + n) x = m x_{2} + n x_{1} .$ Step 5 — divide by $m + n$ (allowed only when $m + n \neq = 0$ ): $x = \frac{m x _{2} + n x _{1}}{m + n} .$ By the same steps on the vertical shadow $\frac{y - y _{1}}{y _{2} - y} = \frac{m}{n}$ : $y = \frac{m y _{2} + n y _{1}}{m + n} .$

Why the topic needs it: the derivation starts from a ratio-equation and must free $x$ from inside the fraction. Cross-multiplication plus these five tidy-up steps is the whole tool.

A

and

B

stand for?

The two given endpoints of the segment,

A = (x_{1}, y_{1})

and

B = (x_{2}, y_{2})

What does the pair

(x, y)

tell you to do from the origin?

Walk

x

steps horizontally, then

y

steps vertically.

Is the

1

x_{1}

a multiplication, a power, or a label?

A label — it just names the point (

A

), nothing else.

What do the bare letters

x, y

(no subscript) stand for?

The coordinates of the splitting point

P = (x, y)

— the unknowns we solve for.

What does writing

A P

(two letters, no operation) mean?

The unsigned length of the rope from

A

P

What extra information does the directed length

\overline{A P}

carry?

A sign telling you the direction of travel; it can be negative.

When does

A P + P B = A B

hold?

Only when

P

lies between

A

and

B

(internal division).

What does the ratio

2 : 1

mean for a segment?

Cut into

3

equal parts; first piece takes

2

, second takes

1

For internal division, what signs must

m

and

n

have?

Both positive (

m > 0

n > 0

Where does

P

land if

n = 0

? And if

m = 0

B

(ratio

m : 0

); on

A

(ratio

0 : n

Rewrite

\frac{a}{b} = \frac{c}{d}

without fractions.

a \cdot d = b \cdot c

(cross-multiply).

Starting from

n (x - x_{1}) = m (x_{2} - x)

, what is

x

x = \frac{m x _{2} + n x _{1}}{m + n}

(expand, gather

x

, factor, divide).

Internal vs external — what's the difference in one word?

Between (internal) vs beyond (external) the segment.

How do you turn the internal formula into the external one?

Replace every

n

- n

: pluses become minuses, giving

\frac{m x _{2} - n x _{1}}{m - n}

What condition must hold to avoid dividing by zero?

m + n \neq = 0

(internal) and

m - n \neq = 0

(external).

The shadow-triangle picture breaks on a vertical line — does the formula?

No: run the same argument on the vertical shadows;

x = x_{1} = x_{2}

and

y

splits the ratio.

What makes the axis shadows split in the same ratio as the rope?

The line's constant slope makes the sub-triangles similar.

Write the

y

-coordinate of the internal division point.

y = \frac{m y _{2} + n y _{1}}{m + n}

The formula

\frac{m x _{2} + n x _{1}}{m + n}

is what kind of average?

A weighted average of

x_{2}

and

x_{1}

Connections

Section formula (Hinglish) — the parent this page prepares you for.
Similar triangles — the geometric engine behind §7.
Slope of a line — constant slope is why the shadow triangles are similar.
Distance formula — measures the lengths $A P$ , $P B$ this page names.
Midpoint formula — the equal-weight special case of §9.
Centroid of a triangle — a $2 : 1$ weighted mix built on the same ideas.
Collinearity of three points — needs "one point divides two others in a ratio".

Foundations — Section formula — internal and external division