Question

the perpendicular AD on the base BC of a triangle ABC intersects BC dt D so that DB=3CD. prove that 2AB²=2AC²+BC²​

Answer 1

The major Mughal emperors were:

Babur (r. 1526-30)

Humayun (r. 1530-56)

Akbar (r. 1556-1605)

Jahangir (r. 1605-27)

Shah Jahan (r. 1627-58)

Aurangzeb (r. 1658-1707)

Answer 2

★Given:-

• The perpendicular AD on the base BC of a triangle ABC intersects BC at D so that DB=3CD.

★To prove:-

• 2AB²=2AC²+BC²

★Solution:-

From ΔACD,

Using pythagoras theorem,

⇒AC² = AD²+DC²

⇒ AD² = AC²-DC²------(1)

From ΔABD,

By pythagoras theorem,

⇒AB²=AD²+DB²

⇒AD² = AB²-DB²------(2)

From equation (1) &(2) ,

⇒AC²-DC²=AB²-DB²------(3)

Given that,

• DB = 3CD

And we have,

• BC = DB + CD

⇒ BC = 3CD +CD

⇒ BC = 4CD

⇒ CD = BC/4

Likewise,

⇒ BC = DB+CD

⇒ BC =DB+(BC/4)

⇒ DB = 3BC/4

Now substituting the value of CD & DB in equation (3),

⇒AC²-DC²=AB²-DB²

⇒AC²-(BC/4)²=AB²-(3BC/4)²

⇒AC²-(BC²/16) = AB²-(9BC²/16)

⇒16AC²-BC² = 16AB²-9BC²

⇒ 16AB²-16AC²=8BC²

⇒ 16(AB²-AC²) = 8BC²

⇒ 2AB²-2AC²=BC²

⇒ 2AB² =2AC²+BC²

Hence proved !

______________