Student Trove
The Proportion Theorem
The Intuition (Plain English)
If you draw a line straight across a triangle so that it is perfectly parallel to the base, it will slice the left side and the right side in exactly the same ratio. If it cuts the left side exactly in half, it will also cut the right side exactly in half. If it cuts the left side into a 1:3 ratio, the right side is also cut into a 1:3 ratio.
The Formal Proof
This is the first of the major Grade 12 proofs. It is a classic "Area" proof, heavily relying on the fact that triangles with the same height and base have the same area.
Given: ΔABC with line DE drawn parallel to BC, intersecting AB at D and AC at E.
R.T.P: AD/DB = AE/EC
Construction: Draw altitudes h (from E to AD) and k (from D to AE). Join BE and CD.
| Statement | Reason |
|---|---|
| 1. Area ΔADE / Area ΔBDE = (½ · AD · h) / (½ · DB · h) = AD/DB | same height h |
| 2. Area ΔADE / Area ΔCDE = (½ · AE · k) / (½ · EC · k) = AE/EC | same height k |
| 3. But Area ΔBDE = Area ΔCDE | same base DE and same height (DE ∥ BC) |
| 4. ∴ Area ΔADE / Area ΔBDE = Area ΔADE / Area ΔCDE | from 3 |
| 5. ∴ AD/DB = AE/EC | from 1 and 2 |
How to apply this in an exam
In Paper 2, you are often given ratios (like 3:2) instead of actual lengths. You must assign a variable (like k or x) to the lengths so you don't confuse ratios for absolute cm values.
In ΔPQR, S lies on PQ and T lies on PR such that ST ∥ QR. Given that PS/SQ = 3/5 and PR = 32 cm, calculate the length of PT.
Step-by-Step Solution:
- Apply the proportion theorem:
PT/TR = PS/SQ (line || one side of ∆) - Substitute the known ratio:
PT/TR = 3/5 - This means the total ratio parts for PR is 3 + 5 = 8 parts.
- Therefore, PT makes up 3/8 of the total length of PR.
- PT = 3/8 × 32
PT = 12 cm.
⚠️ Common Exam Traps
- Forgetting to state WHICH lines are parallel in your reason: The DBE requires you to either write
line || one side of ∆ORprop theorem; ST || QR. If you just write "proportion theorem" without stating which lines are parallel, you lose the reason mark. - Confusing Ratio with Length: If AD/DB = 2/3, it does NOT mean AD = 2cm and DB = 3cm. It means AD = 2k and DB = 3k. Always introduce a variable like k if you plan to do calculations with the raw numbers.