# Basic Proportionality Theorem

How to use the basic proportionality theorem to find the segments divided proportionally by parallel lines: examples and their solutions.

## Theorem

If a line parallel to the side of a triangle
divides the other two sides into four segments,
then the divided segments are proportional.

So the ratio of the left segements, [blue]/[green],
and the ratio of the right segments, [dark blue]/[dark green],
are equal.

This is the basic proportional theorem.
(= Intercept theorem, Thales' theorem)

## Example 1

By the middle line parallel to the bottom side,
two sides of a triangle are divided into four segments.

Then these divided segments are proportional.

The divided segments are [5, 3] and [6, x].

So 5/3 = 6/x.

Solve the proportion.

Then 5x is equal to, 3⋅6, 18.

Proportion

Divide both sides by 5.

Then x = 18/5.

## Extending the Basic Proportionality Theorem

If three parallel lines divide two lines,
then the divided segments are also proportional.

To see why this is true,
extend the two divided lines (dashed line)
and make a triangle.

Then, by the basic proportionality theorem,
the divided segments are proportional.

## Example 2

By three parallel lines,
two lines are divided into four segments.

Then these divided segments are proportional.

The divided segments are [8, 16] and [x, 18].

So 8/16 = x/18.

Cancel 8 and write 1
and reduce 16 to 2.

Solve the proportion.

Then 2x = 18.

Proportion

Divide both sides by 2.

Then x = 9.

## Example 3

Three vertical lines
are all perpendicular to the bottom line.

Then, by the perpendicular transversal theorem,
these three vertical lines are all parallel.

By these three vertical lines,
the upper line and the bottom line
are divided into four segments.

Then these divided segments are proportional.

The divided segments are [11 - x, x] and [6, 4].

So (11 - x)/x = 6/4.

Reduce 6 to 3
and reduce 4 to 2.

Solve the proportion.

Then 2(11 - x) = 3x.

Proportion

2(11 - x) = 22 - 2x

Move 22 to the right side.
And move 3x to the left side.

Then, -2x - 3x, -5x is equal to -22.

Multiply (-) on both sides.

Then 5x = 22.

Divide both sides by 5.

Then x = 22/5.

## Example 3: Another Solution

Let's solve this problem differently,
by setting a different proportion.

Three vertical lines
are all perpendicular to the bottom line.

Then, by the perpendicular transversal theorem,
these three vertical lines are all parallel.

By these three vertical lines,
the upper line and the bottom line
are divided into four segments.

Then these divided segments are proportional.

So the whole segments are also proportional.

The whole segments are 11 and 10.
And the right segments are x and 4.

So 11/x = 10/4.

Just like this proportion,
you can also use the whole segment
with the divided segments
to make a proportion.

Reduce 10 to 5
and reduce 4 to 2.

Solve the proportion.

Then 5x is equal to, 11⋅2, 22.

Proportion

Divide both sides by 5.

Then x = 22/5.

As you can see,
you can get the same answer.