Question

in a right angle 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]