Question-13498

Question Number 13498 by sin (x) last updated on 20/May/17

Commented by ajfour last updated on 20/May/17

if △ABC is equilateral, Area(△ABC)=12×Area(△AED).

$${if}\:\bigtriangleup{ABC}\:{is}\:{equilateral}, \\ $$$${Area}\left(\bigtriangleup{ABC}\right)=\mathrm{12}×{Area}\left(\bigtriangleup{AED}\right). \\ $$

Answered by mrW1 last updated on 21/May/17

since AE=EB ⇒A_(ΔAEC) =A_(ΔBEC) ⇒A_(ΔAEC) =(1/2)A_(ΔABC) since DC=5×AD ⇒A_(ΔCED) =5×A_(ΔAED) ⇒A_(ΔAED) =(1/6)×A_(ΔAEC) =(1/6)×(1/2)A_(ΔABC) ⇒A_(ΔABC) =12×A_(ΔAED) =12×4=48 cm^2 ⇒Answer ?)

$${since}\:{AE}={EB} \\ $$$$\Rightarrow{A}_{\Delta{AEC}} ={A}_{\Delta{BEC}} \\ $$$$\Rightarrow{A}_{\Delta{AEC}} =\frac{\mathrm{1}}{\mathrm{2}}{A}_{\Delta{ABC}} \\ $$$${since}\:{DC}=\mathrm{5}×{AD} \\ $$$$\Rightarrow{A}_{\Delta{CED}} =\mathrm{5}×{A}_{\Delta{AED}} \\ $$$$\Rightarrow{A}_{\Delta{AED}} =\frac{\mathrm{1}}{\mathrm{6}}×{A}_{\Delta{AEC}} =\frac{\mathrm{1}}{\mathrm{6}}×\frac{\mathrm{1}}{\mathrm{2}}{A}_{\Delta{ABC}} \\ $$$$\Rightarrow{A}_{\Delta{ABC}} =\mathrm{12}×{A}_{\Delta{AED}} =\mathrm{12}×\mathrm{4}=\mathrm{48}\:{cm}^{\mathrm{2}} \\ $$$$ \\ $$$$\left.\Rightarrow{Answer}\:?\right) \\ $$

Commented by mrW1 last updated on 21/May/17