Question

In Triangle ABC , if AD is median, show that
$ab^{2} +ac^{2} = 2(ad ^{2} + bd ^{2} )$
​

Answer 1

The rule of the Chalukyas marks an important milestone in the history of South India and a golden age in the history of Karnataka. The political atmosphere in South India shifted from smaller kingdoms to large empires with the ascendancy of Badami Chalukyas. A Southern India-based kingdom took control and consolidated the entire region between the Kaveri and the Narmada rivers. The rise of this empire saw the birth of efficient administration, overseas trade and commerce and the development of new style of architecture called "Chalukyan architecture". Kannada literature, which had enjoyed royal sup

port in the 9th century Rashtrakuta court found eager patronage from the Western Chalukyas in the Jain and Veerashaiva traditions. The 11th century saw the patronage of Telugu literature under the Eastern Chalukyas.

Answer 2

Answer:

AB² + AC² = 2(AD² + BD²).

Step-by-step explanation:

(i)

In ΔAED,

⇒ AD² = AE² + DE²

⇒ AE² = AD² - DE²

(ii)

In ΔAEB,

⇒ AB² = AE² + BE²

           = AD² - DE² + BE²

           = AD² - DE² + (BD + DE)²  {BE = BD + DE}

           = AD² - DE² + BD² + DE² + 2BD * DE - DE²

           = AD² + BD² + 2BD * DE

(iii)

In ΔAEC,

⇒ AC² = AE² + EC²

           = AD² - DE² + EC²

           = AD² - DE² + (DC - DE)²

           = AD² - DE² + DC² + DE² - 2DC * DE

           = AD² + BD² - 2BD * DE {DC = BD}

On solving (ii) & (iii), we get

⇒ AB² + AC² = AD² + BD² + 2BD * DE + AD² + BD² - 2BD * DE

                     = AD² + BD² + AD² + BD²

                     = 2(AD² + BD)²

Hence proved.!