The Proportionality Theorem

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The Proportionality Theorem

A line parallel to one side of a triangle divides the other two sides proportionally. Its converse can establish parallelism. Similar triangles provide the proof structure.

Worked reasoning

$AD/DB=AE/EC$ by proportional division.
$4/6=5/EC$ .
$4EC=30$ , so $EC=7.5$ .

Key idea

Parallel lines in a triangle create corresponding side ratios through similarity.

In $\triangle ABC$ , $DE\parallel BC$ , $AD=4$ , $DB=6$ and $AE=5$ . Find $EC$ .

Which statement best captures the central mathematical idea in The Proportionality Theorem?

Parallel lines in a triangle create corresponding side ratios through similarity.

Any segment inside a triangle divides the other sides proportionally.

A memorised formula is valid even when its conditions are not met.

The diagram or notation replaces the need to justify each mathematical step.

When starting a problem about The Proportionality Theorem, which move is most reliable?

Mark parallel lines and match vertices of the small and large triangles.

Round every value before identifying the mathematical structure.

Choose the longest available formula and substitute immediately.

Assume the desired conclusion and work backwards without checking equivalence.

Which statement is a misconception that must be rejected when working with The Proportionality Theorem?

Any segment inside a triangle divides the other sides proportionally.

Parallel lines in a triangle create corresponding side ratios through similarity.

Mark parallel lines and match vertices of the small and large triangles.

Verify corresponding ratios use matching small-to-large orientation.

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