L4 · The proportionality theorem and the altitude theorem

Learning goal

“I can use the triangle proportionality theorem (a line parallel to one side divides the other two sides proportionally, creating a smaller similar triangle) and the altitude theorem (the altitude to the hypotenuse of a right triangle creates two smaller triangles similar to the original and to each other) to set up and solve proportions for unknown lengths.”

Parallel cut — a line segment drawn inside a triangle, parallel to one side and joining the other two sides (the setup of the proportionality theorem).

Proportion — an equality of two ratios, like 4/6 = 6/9 - the working tool of this lesson; solved by cross-multiplication.

Altitude — a segment from a vertex perpendicular to the opposite side; in this lesson, from the right angle of a right triangle to its hypotenuse.

Hypotenuse — the side opposite the right angle, the longest side of a right triangle (review term, load-bearing today).

Worked example

In triangle ABC, point D lies on AB and point E lies on AC, with DE parallel to BC. Given AD = 4, DB = 6, AE = 6, find EC. Step 1: Why similarity holds: angle A is common to triangle ADE and triangle ABC, and DE parallel to BC makes the corresponding angles at D and E equal to those at B and C. So triangle ADE is similar to triangle ABC by AA - the cut creates a smaller copy of the whole. Step 2: Because the smaller triangle is a scaled copy, the sides are divided proportionally: AD/DB = AE/EC. Step 3: Substitute: 4/6 = 6/EC, so 4 x EC = 36 and EC = 9. (Also note AD/AB = 4/10: the scale factor of the small triangle is 2/5.) Say it plainly: parallel lines preserve angles; preserved angles give AA; AA gives proportions. The theorem is the shortcut name for that whole chain.

Check yourself — explain it

The proportionality theorem works because the cut-off triangle is similar to the whole triangle. That similarity comes from...

Paired problem — example still visible

In triangle ABC, DE is parallel to BC with D on AB and E on AC. AD = 3, DB = 9, AE = 2. Find EC -> 3/9 = 2/EC, so EC = 6

Faded completion — last step is yours

"AD/DB = 5/10 and AE = 4: AE/EC = 5/10 gives EC = ___" -> 8

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10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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