Basic Proportionality Theorem

How to use the basic proportionality theorem to find the segments divided proportionally by parallel lines: 3 formulas, 3 examples, and their solutions.

Formula

Formula

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

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

Example 1

Example

Solution

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

Then these divided segments are proportional.

The left segments are
(top) = 5 and (bottom) = 3.

The right segments are
(top) = 6 and (bottom) = x.

So 5/3 = 6/x.

Solve the proportion.

Then 5x = 6⋅3.

6⋅3 = 18

Divide both sides by 5.

Then x = 18/5.

So x = 18/5.

Extended Formula 1

Formula

If three parallel lines divide two lines,
then the divided segments are also proportional:
a/b = a'/b'.

If two lines are not parallel,
two lines will meet at a point.
(dashed line)

Then you can see that
this formula is the extended version
of the basic proportionality theorem.

(If two lines are parallel,
then a = a' and b = b'.)

Example 2

Example

Solution

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

Then these divided segments are proportional.

The left segments are
(top) = 8 and (bottom) = 16.

The right segments are
(top) = x and (bottom) = 18.

So 8/16 = x/18.

8/16 = 1/2

1/2 = x/18

Solve the proportion.

Then 2x = 18.

Divide both sides by 2.

Then x = 9.

So x = 9.

Extended Formula 2

Formula

If the whole segment is given,
use this extended formula:
(a + b)/b = (a' + b')/b'.

Example 3

Example

Solution

Three vertical lines
are all perpendicular to the bottom line.

Then the vertical lines are all parallel.
(The right angles are
corresponding angles in parallel lines.)

The whole length of the top segment is 11.

So write the whole length of the bottom segment:
6 + 4 = 10.

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

The top segments are
(whole) = 11 and (right) = x.

The bottom segments are
(whole) = 10 and (right) = 6.

So 11/x = 10/6.

10/4 = 5/2

11/x = 5/2

Solve the proportion.

Then 5x = 11⋅2.

11⋅2 = 22

5x = 22

Divide both sides by 5.

Then x = 22/5.

So x = 22/5.