Prove BPT

1.	Prove BPT
Answer» Basic Proportionality Theorem (Thales theorem):\xa0If a line is drawn parallel to one side of a triangle intersecting other two sides, then it divides the two sides in the same ratio.\xa0\xa0\tIn ∆ABC , if DE \|\| BC and intersects AB in D and AC in E then\xa0 AD AE ---- = ------ DB EC\tProof on Thales theorem :\xa0If a line is drawn parallel to one side of a triangle and it intersects the other two sides at two distinct points then it divides the two sides in the same ratio.Given :\xa0In ∆ABC , DE \|\| BC and intersects AB in D and AC in E.\xa0Prove that :\xa0AD / DB = AE / EC\xa0Construction :\xa0Join BC,CD and draw EF ┴ BA and DG ┴ CA.\xa0\xa0\xa0\t Statements Reasons 1) EF ┴ BA1) Construction2) EF is the height of ∆ADE and ∆DBE2) Definition of perpendicular3)Area(∆ADE) = (AD .EF)/23)Area = (Base .height)/24)Area(∆DBE) =(DB.EF)/24) Area = (Base .height)/25)(Area(∆ADE))/(Area(∆DBE)) = AD/DB5) Divide (4) by (5)6) (Area(∆ADE))/(Area(∆DEC)) = AE/EC6) Same as above7) ∆DBE ~∆DEC7) Both the ∆s are on the same base and\xa0 between the same \|\| lines.8) Area(∆DBE)=area(∆DEC)8) If the two triangles are similar their\xa0 areas are equal9) AD/DB =AE/EC9) From (5) and (6) and (7)\t

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