Question

In ∆ABC, ∠B = 90° and D is mid-point of BC. Prove that :
AC² = AD² + 3CD².

Answer 1

AC² = AD² + 3CD²

Explanation:

Given: In ∆ABC, ∠B = 90° and D is mid-point of BC.

To Prove : AC² = AD² + 3CD².

Proof:

In △ABD, AD² = AB² + BD²

AB² = AD² - BD² ---------(1)

In △ABC, AC² = AB² + BC²

AB² = AC² - BC² ---------(2)

Equation (1) = (2), so we get:

AD² - BD² = AC² - BC²

AD² - BD² = AC² - (BD + DC)² ----------------(as D is the midpoint of BC)

AD² - BD² = AC² - BD²- DC²- 2*BD*DC

AD² = AC² - DC² - 2DC² ---------------------- (DC = BD)

AD² = AC² - 3DC²

AC² = AD² + 3CD²

Hence proved.