Question

In an equilateral triangle ABC, if ADIBC
then​

Answer 1

$\huge\star\:\:{\orange{\underline{\purple{\mathcal{Solution}}}}}$

△ABC is a equilateral triangle.

Thus, AB=BC=AC and∠ABC=∠BAC=∠ACB=60

∠ADB=∠ADC=90

In △ABD,∠BAD+∠ABD+∠ADB=180

⇒∠BAD=180−90−60=30

Similarly for △ACD,∠CAD+∠ACD+∠ADC=180

⇒∠CAD=180−90−60=30

Now, In △ABD and △ACD

AB=AC

∠BAD=∠CAD

AD=AD(Common side)

Thus, △ABD and △ACD are congruent.(SAS)

$\underline\bold\blue {Therefore: -}$

$BD =DC = \frac{1}{2} BC= \frac{1}{2} AB$

Now, △ABD is a right angled triangle.

$\underline\bold\red {Therefore: -}$

${ AB}^{2} = {BD}^{2} + {AD}^{2}$

$⇒ {AB}^{2} = { (\frac{1}{2})AB }^{2} + {AD}^{2}$

$⇒ {AB}^{2} = \frac{ {AB}^{2} }{4} + {AD}^{2}$

$⇒ {AB}^{2} - \frac{ {AB}^{2} }{4} = {AD}^{2}$

$⇒ \frac{ {4 AB}^{2} - { AB}^{2} }{4} = {AD}^{2}$

$⇒ 3 {AB}^{2} = 4 {AD}^{2}$

Hence verified ✔