Question

In an equilateral triangle ABC , D is the point on side BC such that BD =1/3 BC.then AD²/AB² is
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Answer 1

Given:

△ABC is an Equilateral triangle such that,

AB=BC=AC

Construction:

Draw an Altitude AE such that,

E lies on BC and

AE perpendicular to BC

In △AEB

By Pythagoras Theorem,

AB² =AE²+EB².....(1)

In △AED

By Pythagoras Theorem,

AD²=AE²+ED².....(2)

From 1) and 2)

AD²=ED²+AB²−EB² .....(3)

Since, EB = 1/2BC = AB/2 = 3AB/6

and ED = BC/6 = AB/6

∴ AD² = AB²/36 + AB²-9AB²/36

∴ 9AD²=7AB²