Question

In a right angled triangle, one acute angle is double the other. Prove that the hypotenuse is double the smallest side. 

Answer 1

Let ∆ABC is a right triangle such that ∠B = 90⁰ and ∠ACB = 2∠CAB, then we have to prove AC = 2BC. we produce CB to D such that BD = CB and join AD.

Proof : In ∆ABD and ∆ABC we have 

BD = BC [By construction] 

 AB = AB [Common] 

 ∠ABD = ∠ABC = 90⁰ 

∴ By SAS criterion of congruence we get 

 ∆ABD ≅ ∆ABC 

⇒ AD = AC and ∠DAB = ∠CAB [By cpctc] 

⇒ AD = AC and ∠DAB = x [∴∠CAB = x] 

 Now, ∠DAC = ∠DAB + ∠CAB = x + x = 2x 

∴ ∠DAC = ∠ACD ⇒ DC = AD [Side Opposite to equal angles] 

⇒ 2BC = AD [∵DC = 2BC]

⇒ 2BC = AC [AD = AC]