In fig., ABC is a triangle in which ∠ABC < 90° and AD ⊥ BC. Prove that AC² = AB² + BC²

Proof: In △ADC
By Pythagoras Theorem, AC² = AD² + DC² (1)
In △ABD
By Pythagoras Theorem, AB² = AD² + BD² (2)
Subtracting (1) and (2) we get,
AC² - AB² = DC² - BD²
∴ AC² - AB² = DC² - (BC - DC)²
∴ AC² - AB² = 2DC.BC - BC²
∴ AC² - AB² = 2(BC - BD)BC - BC²
∴ AC² - AB² = -2DB.BC + 2BC² - BC²
∴ AC² = AB² + BC² - 2BC.BD

Eduprobe

Question:

In fig., ABC is a triangle in which ∠ABC < 90° and AD ⊥ BC. Prove that AC² = AB² + BC² - 2BC.BD.

Solution: