Question

If triangle ABC is an equilateral triangle such that AD is perpendicular to BC then (AD)2 =k (BE)2 find the value of k

Answer 1

K = 3  if triangle ABC is an equilateral triangle such that ad perpendicular BC then ad²= K(BD)²

Step-by-step explanation:

Correction  BE is BD

ABC is an equilateral triangle

=> AB = BC = AC

AD ⊥ BC

=> ΔABD ≅ ΔACD

as AB = AC

    AD = AD  ( common)

   ∠ADB = ∠ADC = 90°

=> BD = CD  = BC/2

=> BC = 2 BD

=> AB = 2BD

AD² = AB² - BD²

=>AD² = (2BD)² - BD²

=> AD² = 3BD²

K = 3

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Answer 2

The value of K is 3

Step-by-step explanation:

GIVEN  :  ABC is an equilateral triangle

AD ⊥ BC

proof : since , ABC is an equilateral triangle

AB = BC = CA

in triangle ABD and ACD

AB = AC

AD = DA (common)

$\angle ADB = \angle ADC = 90^\circ$

therefore ,

$\bigtriangleup ABD \cong \bigtriangleup ACD$

(by SAS)

therefore ,

BD = CD

$BD = \frac{BC}{2}$

BC = 2BD

AB = BC

AB = 2 BD

now using Pythagoras theoren in triangle ACD

AD² = AB² - BD²

AD² = (2BD)² - BD²

AD² = 4BD² - BD²

AD² = 3BD²

hence , The value of K is 3

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