الأسبوع 9 · المستقيمات المتوازية والأجزاء المتناسبة: نظرية التناسب في المثلث وعكسها ونظرية تاليس · 3 دروس
1
A line parallel to one side of a triangle divides the other two sides in proportion: AD/DB = AE/EC, inside and outside the triangle
Mathematics Sec 1 Term 1, Unit 3, Lesson 1
GoalUse the theorem that a line parallel to a side divides the other two sides in the same ratio.
New words
proportional parts · pieces of two segments that have the same ratiodivides in the ratio · cuts a segment into two parts whose ratio is given
Explanation
Theorem: if a line DE parallel to BC meets AB at D and AC at E, then AD/DB = AE/EC. Proof: △ADE ~ △ABC, so AB/AD = AC/AE. Subtract 1 from both sides: AB/AD − 1 = (AB − AD)/AD = DB/AD, and in the same way AC/AE − 1 = EC/AE. So DB/AD = EC/AE, and turning both fractions upside down gives AD/DB = AE/EC. We also keep AD/AB = AE/AC and DE/BC = AD/AB. The theorem still holds when D and E lie on the extensions of AB and AC beyond A, B or C: then AD/AB = AE/AC with all lengths measured from A. Be careful which pieces form the ratio: AD/DB compares two parts of one side, while AD/AB compares a part with the whole side. Both are true, but they are different ratios and cannot be mixed.
Fact
Written as
part to part (same side)
AD/DB = AE/EC
part to whole
AD/AB = AE/AC
all three (with the base)
AD/AB = AE/AC = DE/BC
Worked examples
DE ∥ BC with D on AB and E on AC: AD = 6, DB = 4 and AE = 9. Find EC and AC. AD/DB = 6/4 = 3/2 = AE/EC = 9/EC, so EC = 6 and AC = 9 + 6 = 15.
DE ∥ BC with AB = 20, AD = 8 and AC = 25. Find AE and EC. AD/AB = 8/20 = 2/5 = AE/AC, so AE = 10 and EC = 15. Check: AD/DB = 8/12 = 2/3 and AE/EC = 10/15 = 2/3 ✓.
Outside the triangle. D is on AB extended beyond B and E is on AC extended beyond C with BC ∥ DE. AB = 8, BD = 4 and AC = 10. Then AD = 12 and AB/AD = AC/AE gives 8/12 = 10/AE, so AE = 15 and CE = 5.
Class activity
Ratio hunt. In the figure of the first example, write three different true ratios that equal AD/DB or AD/AB, and decide which pairs may be set equal.
Practice
1
DE ∥ BC, D on AB, E on AC, AD = 4, DB = 8, AE = 3. Find EC.
4/8 = 3/EC, so EC = 6.
2
DE ∥ BC, AD = 5, AB = 15 and EC = 14. Find AE and AC.
DB = 10, AD/DB = 1/2 = AE/14, so AE = 7 and AC = 21.
3
DE ∥ BC with AD = x, DB = 3, AE = 8 and EC = 12. Find x.
x/3 = 8/12 = 2/3, so x = 2.
4
D is the midpoint of AB and DE ∥ BC. If AC = 18, find AE.
AD/DB = 1 = AE/EC, so E is the midpoint and AE = 9.
5
Hourglass: lines BD and CE meet at A between the parallels BC and DE. AB = 6, AD = 15 and AC = 4. Find AE.
