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Basic Proportionality Theorem (with proof) sheet
5-mark answer frame

State and prove the Basic Proportionality Theorem.

How to lay out the answer. Follow the frame, then write it in your own words.

  1. 1 · Write what is given and what is to be found. Statement: a line drawn parallel to one side of a triangle, cutting the other two sides at distinct points, divides them in the same ratio. Given: ΔABC, DE ∥ BC, D on AB, E on AC. To prove: AD/DB = AE/EC.
  2. 2 · Draw the figure. Draw ΔABC with DE ∥ BC. Join BE and CD. Draw EN ⊥ AB and DM ⊥ AC. Label all points.
  3. 3 · Write the formula or theorem. Construction: join BE and CD; draw DM ⊥ AC and EN ⊥ AB. Use area of a triangle = ½ × base × height.
  4. 4 · First ratio of areas.
    • ar(ADE) = ½ × AD × EN and ar(BDE) = ½ × DB × EN
    • ar(ADE)/ar(BDE) = AD/DB ... (1)
  5. 5 · Second ratio of areas.
    • ar(ADE) = ½ × AE × DM and ar(DEC) = ½ × EC × DM
    • ar(ADE)/ar(DEC) = AE/EC ... (2)
  6. 6 · Equal areas. ΔBDE and ΔDEC are on the same base DE and between the same parallels BC and DE, so ar(BDE) = ar(DEC) ... (3)
  7. 7 · Check and box the answer. From (1), (2) and (3), AD/DB = AE/EC. Hence proved.

Scoring tip: The figure, the construction and the 'same base, same parallels' reason carry most of the marks. Never skip the reason for step (3).