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Sep 1, 2014

# Prove that 9AD2 = 7AB2

In an equilateral Î” ABC, the side BC is trisected at D. Prove that 9AD2 = 7AB2

NextGurukul Guest User

Member since Apr 1, 2017

In this solution triangle ABC and ACE is taken in place of ABC, ABE should be taken.

Raghunath Reddy

Member since Apr 11, 2014

Sol: Given:   In an equilateral triangle Î”ABC. The side BC is trisected at D such that BD = (1/3) BC. To prove:  9AD2  = 7AB2 Construction:  Draw AE âŠ¥ BC. Proof : In a Î”ABC and Î”ACE AB = AC ( Given) AE = AE ( common) âˆ AEB = âˆ AEC = 90Â° âˆ´ Î”ABC â‰… Î”ACE ( For RHS criterion) BE = EC (By C.P.C.T) BE = EC = BC / 2 In a right angled triangle ADE AD2 = AE2 + DE2 ---------(1) In a right angled triangle ABE AB2 = AE2 + BE2 ---------(2) From equ (1) and (2) we obtain â‡’ AD2  - AB2 =  DE2 - BE2 . â‡’ AD2  - AB2 = (BE â€“ BD)2 - BE2 . â‡’ AD2  - AB2 = (BC / 2 â€“ BC/3)2 â€“ (BC/2)2 â‡’ AD2  - AB2 = ((3BC â€“ 2BC)/6)2 â€“ (BC/2)2 â‡’ AD2  - AB2 = BC2 / 36 â€“ BC2 / 4 ( In a equilateral triangle Î”ABC, AB = BC = CA) â‡’ AD2 = AB2 + AB2 / 36 â€“ AB2 / 4 â‡’ AD2 = (36AB2 + AB2â€“ 9AB2) / 36 â‡’ AD2 = (28AB2) / 36 â‡’ AD2 = (7AB2) / 9 9AD2 = 7AB2 .

Syeda

Member since Jan 25, 2017

Answer. Given:   In an equilateral triangle Î”ABC. The side BC is trisected at D such that BD = (1/3) BC. To prove:  9AD2  = 7AB2 Construction:  Draw AE âŠ¥ BC. Proof : In a Î”ABC and Î”ACE AB = AC ( Given) AE = AE ( common) âˆ AEB = âˆ AEC = 90Â° âˆ´ Î”ABC â‰… Î”ACE ( For RHS criterion) BE = EC (By C.P.C.T) BE = EC = BC / 2 In a right angled triangle ADE AD2 = AE2 + DE2 ---------(1) In a right angled triangle ABE AB2 = AE2 + BE2 ---------(2) From equ (1) and (2) we obtain â‡’ AD2  - AB2 =  DE2 - BE2 . â‡’ AD2  - AB2 = (BE â€“ BD)2 - BE2 . â‡’ AD2  - AB2 = (BC / 2 â€“ BC/3)2 â€“ (BC/2)2 â‡’ AD2  - AB2 = ((3BC â€“ 2BC)/6)2 â€“ (BC/2)2 â‡’ AD2  - AB2 = BC2 / 36 â€“ BC2 / 4 ( In a equilateral triangle Î”ABC, AB = BC = CA) â‡’ AD2 = AB2 + AB2 / 36 â€“ AB2 / 4 â‡’ AD2 = (36AB2 + AB2â€“ 9AB2) / 36 â‡’ AD2 = (28AB2) / 36 â‡’ AD2 = (7AB2) / 9 9AD2 = 7AB2 . SME Approved
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