Question : Let ABC be an equilateral triangle and AD perpendicular to BC, Then AB2 + BC2 + CA2 =?
Option 1: 2AD2
Option 2: 3AD2
Option 3: 4AD2
Option 4: 5AD2
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Correct Answer: 4AD 2
Solution :
Here ABC is an equilateral triangle with side length $a$ and AD $\perp$ BC.
So, BD = CD = $\frac{a}{2}$
In $\triangle$ABD, AB
2
= BD
2
+ AD
2
⇒ $a^2=(\frac{a}{2})^2+$ AD
2
$\therefore$ AD = $\frac{\sqrt3}{2}a$ = $\frac{\sqrt3}{2}$AB = $\frac{\sqrt3}{2}$BC = $\frac{\sqrt3}{2}$AC [$\because$ AB = BC = AC = $a$]
⇒ AB = $\frac{2}{\sqrt3}$ AD, BC = $\frac{2}{\sqrt3}$ AD, and AC = $\frac{2}{\sqrt3}$ AD
$\therefore$ AB
2
+ BC
2
+ AC
2
= AD
2
($\frac{4}{3}+\frac{4}{3}+\frac{4}{3}$) = 4AD
2
Hence, the correct answer is 4AD
2
.
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