Question

if the angle of a triangle are in the ratio 1:1:2 then find the ratios of the perimeter of the triangle to the longest side.

Answer 1

let the angles be x , x and 2x

we know

x + x + 2x = 180°

4x = 180°

x = 45°

now the angles are -

45° , 45° and 2(45) = 90°

so it is an isoceles right angles triangle

In an isosceles right triangle, the equal sides make the right angle.

They have the ratio of equality, 1 : 1.
To find the ratio

let the length of the hypotenuse be h,

according to the Pythagorean theorem,
h^2 = 1^2 + 1^2 = 2.

Therefore,
h = root(2)

the ratio is - 1:1:root(2)