In the given figure, ΔABC is an obtuse triangle, obtuse angled at B. If AD⊥CB , then prove that AC² = AB² + BC² + 2BC.BD

Solution:

From the right angled triangle ADC,

Using pythagoras theorem, AD² + DC² = AC² --- (1)

From the figure,

DC = DB +BC --- (2)

Substitute DC from (2) in equation (1)

AD² + (BD + BC)² = AC²

AC² = AD² + BD² + BC² + 2BD.BC --- (3)

Also from right angled triangle ADB,

AD² + BD² = AB² (From pythagoras theorem) --- (4)

Substitute (4) in equation (3)

AC² = AB² + BC² + 2BD.BC

In the given figure, ΔABC is an obtuse triangle, obtuse angled at B. If AD⊥CB , then prove that AC² = AB² + BC² + 2BC.BD ?

Summary:

Given ΔABC is an obtuse triangle, obtuse angled at B and AD⊥CB, then AC² = AB² + BC² + 2BC.BD.