Question

THEOREM 6.1 if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points the other two sides are are divided in the same ratio

Answer 1

• draw a line DE parallel to BC of triangle ABC.
• Given: BC||DE.
• To prove : AD /DB= AE/EC.
• Construction: join DC and BE.draw DN perpendicular to AE and EM perpendicular to AD.
• proof: in ∆ADE, ar( ∆ADE ) = 1/2× DN× AE---(1) and also ar(∆ADE) = 1/2×EM×AD---(2) ar(∆DBE)= 1/2×DB× EM---(3) ar(∆DCE)=1/2×CE×DN.-----(4)
• on dividing (1)and (4): ar(∆ADE) / ar(∆DCE)= 1/2×DN×AE/1/2× CE×DN= AE/CE---(5)
• on dividing (2)and(3): ar(∆ADE)/ar(∆DBE)=AD/DB ---(6)
• ar(∆DBE)=ar(∆DCE) (both ∆ are on same base and same between same parallel lines
• (5)=(6),hence AE/CE=AD/DB.