Question

In a right ∆ABC right-angled at C, if D is the mid-point of BC, prove that BC² = 4(AD² − AC²).

Answer 1

It is given that ∆ABC is a right-angled at C and D is the mid-point of BC.

In the right angled triangle ADC, we will use Pythagoras theorem,

AD2 = DC2 + AC2             ..........(1)

Since D is the midpoint of BC, we have

`"DC"/="BC"/2`

Substituting `"DC"/="BC"/2` in equation (1), we get

`"AD"^2=("BC"/2)^2+"AC"^2`

`"AD"^2="BC"^2/4+"AC"^2`

4AD² = BC² + 4AC²

BC² = 4AD² - 4AC²

BC² = 4(AD² - AC²)