Question

state and prove thales theorm

Answer 1

Given : In ∆ABC , DE || BC and intersects AB in D and AC in E.

Prove that : AD / DB = AE / EC

Construction : Join BC,CD and draw EF ┴ BA and DG ┴ CA.

Statements
Reasons
1) EF ┴ BA
1) Construction
2) EF is the height of ∆ADE and ∆DBE
2) Definition of perpendicular
3)Area(∆ADE) = (AD .EF)/2
3)Area = (Base .height)/2
4)Area(∆DBE) =(DB.EF)/2
4) Area = (Base .height)/2
5)(Area(∆ADE))/(Area(∆DBE)) = AD/DB
5) Divide (4) by (5)
6) (Area(∆ADE))/(Area(∆DEC)) = AE/EC
6) Same as above
7) ∆DBE ~∆DEC
7) Both the ∆s are on the same base and
between the same || lines.
8) Area(∆DBE)=area(∆DEC)
8) If the two triangles are similar their
areas are equal
9) AD/DB =AE/EC
9) From (5) and (6) and (7)