# In a ∆ABC, AD ⊥ BC and AD2 = BC × CD. Prove ∆ABC is a right triangle - Mathematics

Sum

In a ∆ABC, AD ⊥ BC and AD2 = BC × CD. Prove ∆ABC is a right triangle

#### Solution

In right triangles ADB and ADC, we have

AB^2 = AD^2 + BD^2 ….(i)

and, AC^2 = AD^2 + DC^2 ….(ii)

Adding (i) and (ii), we get

AB^2 + AC^2 = 2 AD^2 × BD^2 + DC^2

⇒ AB^2 + AC^2 = 2BD × CD + BD^2 + DC^2 [∵ AD2 = BD × CD

⇒ AB^2 + AC^2 = (BD + CD)^2 = BC^2

Thus, in ∆ABC, we have

AB^2 = AC^2 + BC^2

Hence, ∆ABC, is a right triangle right-angled at A.

Concept: Pythagoras Theorem
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