Question

In an equilateral triangle ABC, a point D is taken on base BC such that BD:DC=2:1. Prove that 9ADSquar=7ABSquar
$prove \: that\: 9{ad}^{2} = 7 {ab}^{2}$

Answer 1

Given: ΔABC is an equilateral triangle. D is point on BC such that BD =BC.

To prove: 9 AD2 = 7 AB2

Construction: Draw AE ⊥ BC.

Proof ;-

Considering on Triangles which are given below;-


In a ΔABC and ΔACE

AB = AC ( given)

AE = AE (common)

∠AEB = ∠AEC = (Right angle)


∴ ΔABC ≅ ΔACE


By RHS Creition
∴ ΔABC ≅ ΔACE

Again,

BE = EC (By C.P.C.T)

BE = EC = BC 2

In a right angled ΔADE

AD2 = AE2 + DE2 ---(1)

In a right angled ΔABE

AB2 = AE2 + BE2 ---(2)

From equation (1) and (2) ;

 =) AD2  - AB2 =  DE2 - BE2 .

 =) AD2  - AB2 = (BE – BD)2 - BE2 .

 = ) AD2  - AB2 = (BC / 2 – BC/3)2 – (BC/2)2 

 = AD2  - AB2 = ((3BC – 2BC)/6)2 – (BC/2)2 

 = AD2  - AB2 = BC2 / 36 – BC2 / 4


( In a equilateral triangle, All sides are equal to each other)

AB = BC = AC

 = ) AD2 = AB2 + AB2 / 36 – AB2 / 4

 = )AD2 = (36AB2 + AB2– 9AB2) / 36

 = ) AD2 = (28AB2) / 36



 =) AD2 = (7AB2) / 9

 = ) 9AD2 = 7AB2 ‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎
‎
‎
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‎Hence, 9AD2 = 7AB2 

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Answer 2

Answer:

Step-by-step explanation:

As triangle ABC is equilateral BC=AB

Given BD:DC=2:1

BD=2\3BC =2\3AB

Draw AE PERP BC, then E is the mid point of BC,

so BE=1\2 =1\2AB

ED=BD-BE=2\3AB -1\2AB=1\6AB

In triangleABE, angleAEB=90 degree,

AB square=AE square+BE square.... 1

In triangle, angle AED=90 degree,

AD square=AE square + ED square....... 2

Subtracting 2 from 1 we get,

AB square -AD square=BE square-ED square=1\4 AB square - 1\4 AB square = 2\9 AB square.

AD square=AB square -2\9 AB square=7\9 AB square

9 AD square= 7AB square