Question

In right-angled triangle ABC is which ∠C = 90°, if D is the mid-point of BC, prove that AB² = 4 AD² − 3AC².

Answer 1

SOLUTION :

Given :  

∠C = 90° and D is the mid-point of BC.

In ∆ ACD,  

AD² = AC² + CD²  

[By using  Pythagoras theorem]

CD² = AD² - AC² ……….(1)

In ∆ACB,

AB² = AC² + BC²

[By using Pythagoras theorem]

AB² = AC² + (2CD)²

[D is the mid-point of BC]

AB² = AC² + 4CD²

AB² = AC² +4(AD² - AC²)

[from eq 1]

AB² = AC² + 4AD² - 4AC²

AB² = 4AD² - 4AC² + AC²

AB² = 4AD² - 3AC²

HOPE THIS ANSWER WILL HELP YOU…

Answer 2

Answer:

Step-by-step explanation:

Given :-

ΔABC is a right-angled triangle, with ∠C = 90°.

D is the midpoint of BC.

To Prove :-

AB² = 4 AD² - 3AC²

Solution :-

In ΔABC, By applying Pythagoras' theorem, we get

⇒ AB² = AC² + BC²

⇒ AB² = AC² + (2 CD)²

⇒ AB² = AC² + 4 CD²  ....(i)

Similarly, In ΔADC

⇒ AD² = AC² + CD²

⇒ CD² = AD² - AC² .... (ii)

Solving Eq (i) and (ii), we get

⇒ AB² = AC² + 4 (AD² - AC²)

⇒ AB² = AC² + 4 AD² - 4 AC²

⇒ AB² = 4 AD² - 3 AC²

Hence, Proved.