Given a ΔABC in which ∠B=90∘ and AB=√3BC. Prove that ∠C=60∘.
Answer:
- Let D be the midpoint of the hypotenuse AC.
Join BD. - Now, we have AC2=AB2+BC2[ By pythagoras' theorem ]⟹AC2=(√3BC)2+BC2[∵ AB = √3 BC (given) ]⟹AC2=4BC2⟹AC=2BC⟹2CD=2BC[∵ D is the midpoint of AC]⟹CD=BC… (i)
- We know that the midpoint of the hypotenuse of a right triangle is equidistant from the vertices. ∴ BD=CD… (ii) From (i) and (ii), we get BC=BD=CD Therefore, ΔBCD is equilateral and hence ∠C=60∘.