Question

In an equilateral triangle ABC, D is a point on the side BC such that.BD =1/3 BC Prove that 9AD^2= 7AB^2
bhi tell​

Answer 1

Answer:

Given: An equilateral triangle ABC and D be a point on BC such that BD=1/3 BC. To Prove: 9AD^2 = 7AB^2 Construction: Draw AE ⊥ BC. Proof: ∆ABC is an equilateral triangle and AEBC= BE = EC

In ADP,

AD^2=AP^2+DP^2

AD^2=AP^2+(BP-BD)^2

AD^2=AP^2+BP^2+BD^2 -2(BP)(BD)

AD^2=AB^2+(1/3 BC)^2-2(BC/2)(BC/3)

AD^2=7/9AB^2(since BC=AB)

9AD^2=7AB^2

Answer 2

Step-by-step explanation:

➡ Given :-

→ A ∆ABC in which AB = BC = CA and D is a point on BC such that BD = ⅓BC.

➡ To prove :-

→ 9AD² = 7AB² .

➡ Construction :-

→ Draw AL ⊥ BC .

➡ Proof :-

In right triangles ALB and ALC, we have

AB = AC ( given ) and AL = AL ( common )

∴ ∆ALB ≅ ∆ ALC [ By RHS axiom ] .

So, BL = CL .

Thus, BD = ⅓BC and BL = ½BC .

In ∆ALB, ∠ALB = 90° .

∴ AB² = AL² + BL² .......(1) [ by Pythagoras' theorem ] .

In ∆ALD , ∠ALD = 90° .

∴ AD² = AL² + DL² . [ by Pythagoras' theorem ] .

⇒ AD² = AL² + ( BL - BD )² .

⇒ AD² = AL² + BL² + BD² - 2BL.BD .

⇒ AD² = ( AL² + BL² ) + BD² - 2BL.BD .

⇒ AD² = AB² + BD² - 2BL.BD. [ using (1) ]

⇒ AD² = BC² + ( ⅓BC )² - 2( ½BC ). ⅓BC .

[ ∵ AB = BC, BD = ⅓BC and BL = ½BC ] .

⇒ AD² = BC² + 1/9BC² - ⅓BC² .

⇒ AD² = 7/9BC² .

⇒ AD² = 7/9AB² [ ∵ BC = AB ] .

$\huge \green{ \boxed{ \sf \therefore 9AD^{2} = 7AB^{2}. }}$

Hence, it is proved.