Question

Abc is an isolates triangle with ac = bc if ab2 =2ac prove that abc is right angle

Answer 1

Firstly assuming that the triangle ABC is a right triangle with right angled at C.

So that

AB^2 = AC^2 + BC^2

=> AB^2 = 2AC^2 ---> (1)

Given that AB^2 = 2AC ---> (2)

From (1) and (2),

2AC^2 = 2AC

=> AC^2 = AC

=> AC^2 - AC = 0

=> AC(AC - 1) = 0

Therefore, AC = 1 ; AC = 0 (not possible)

From (2),

AB^2 = 2AC

=> AB^2 = 2 × 1

=> AB^2 = 2

=> AB = 2^(1/2) [Square root of 2]

So the ratio of sides,

=> AC : BC : AB

=> 1 : 1 : 2^(1/2)

All triangles with this ratio of sides have the angles in the ratio 45 : 45 : 90.

So there's such a possible isosceles right triangle which satisfy the property given in the question.

Hence our assumption is proved!